surface area of pyramid problem solving

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Last modified on August 3rd, 2023

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Surface area of a pyramid.

The surface area, or total surface area (TSA), of a pyramid, is the entire space occupied by its flat faces. The surface area is measured in square units such as m 2 , cm 2 , mm 2 , or in 2 .

The general formula is:

Surface Area (SA) = ${B+\dfrac{1}{2}Ps}$ , here B = base area, P = base perimeter, s = slant height,

Also  ${\dfrac{1}{2}Ps}$  = lateral surface area ( LSA )  

∴  SA  = B +  LSA

However, there are specific formulas to calculate the surface area of different pyramids. They are given below:

Let us solve some examples involving the above formulas.

Solved Examples

Calculate the surface area of a triangular pyramid with a base area of 24 cm 2 , a base perimeter of 12 cm, and a slant height of 16 cm.

As we know, Surface Area (SA) = ${B+\dfrac{1}{2}Ps}$, here B = 24 cm 2 , P = 12 cm, s = 16 cm ∴ SA = ${24+\dfrac{1}{2}\times 12\times 16}$ = 120 cm 2

Finding the surface area of a triangular pyramid when BASE , BASE HEIGHT , and SLANT HEIGHT are known.

Find the total surface area of a regular triangular pyramid with a slant height of 10 cm, base of 6 cm, and a base height of 5.2 cm.

As we know, Total Surface Area ( TSA ) = ${\dfrac{1}{2}bH+\dfrac{3}{2}bs}$, here b = 6 cm, H = 5.2 cm, s = 10 cm ∴ TSA = ${\dfrac{1}{2}\times 6\times 5.2+\dfrac{3}{2}\times 6\times 10}$  = 285.6 cm 2

Find the lateral and total surface area of a square pyramid with a base of 6 cm and a slant height of 7.3 cm.

As we know, Lateral Surface Area ( LSA ) = 2bs, here b = 6 cm, s = 7.3 cm ∴ LSA = 2 × 6 × 7.3 = 87.6 cm 2 Total Surface Area ( TSA ) = b 2 + LSA , here b = 6 cm, LSA = 87.6 cm 2 ∴ TSA = 6 2 + 87.6 = 123.6 cm 2

Find the total surface area of a rectangular pyramid with a base of 7 cm and 9 cm, and a height of 11 cm.

As we know, Total Surface Area ( TSA ) = ${lw+\dfrac{1}{2}w\sqrt{4h^{2}+l^{2}}+\dfrac{1}{2}l\sqrt{4h^{2}+w^{2}}}$, here l = 9 cm, w = 7 cm, h = 11 cm ∴ TSA = ${9\times 8+\dfrac{1}{2}\times 7\sqrt{4\times 11^{2}+9^{2}}+\dfrac{1}{2}\times 9\sqrt{4\times 11^{2}+7^{2}}}$ = 250.08 cm 2

Find the total surface area of a pentagonal pyramid with a base of 3 cm, apothem of 2.06 cm, and a slant height of 4 cm.

As we know, Total Surface Area ( TSA ) = ${\dfrac{5}{2}b\left( a+s\right)}$, here b = 3 cm, a = 2.06 cm, s = 4 cm ∴ TSA = ${\dfrac{5}{2}\times 3\times \left( 2.06+4\right)}$ = 45.45 cm 2

Find the total surface area of a hexagonal pyramid with a base of 4 cm, apothem of 3.46 cm, and a slant height of 13 cm.

As we know, Total Surface Area ( TSA ) = 3ab + 3bs , here a = 3.46 cm, b = 4 cm, s = 13 cm ∴ TSA = 3 × 3.46 × 4 + 3 × 4 × 13 = 197.52 cm 2

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• Problems on Pyramid

Solved word problems on pyramid are shown below using step-by-step explanation with the help of the exact diagram in finding surface area and volume of a pyramid.

Worked-out problems on pyramid: 1. The base of a right pyramid is a square of side 24 cm. and its height is 16 cm.

(i) the area of its slant surface

(ii) area of its whole surface and

(iii) its volume. 

Let, the square WXYZ be the base of the right pyramid and its diagonals WY and XZ intersect at O. If  OP  be perpendicular to the plane of the square at O, then  OP  is the height of the pyramid.

Draw  OE  ┴  WX Then, E is the mid - point of  WX . 

By question,  OP  = 16 cm. and  WX  = 24 cm.  Therefore,  OE  =  EX  = 1/2 ∙  WX  = 12 cm Clearly,  PE  is the slant height of the pyramid.  Since  OP  ┴  OE , hence from ∆ POE we get,  PE² = OP² + OE² 

or,PE² = 16² + 12² 

or, PE² = 256 + 144 

or, PE² = 400

PE  = √400

Therefore,  PE  = 20.  Therefore, (i) the required area of slant surface of the right pyramid

= 1/2 × perimeter of the base × slant height. 

= 1/2 × 4 × 24× 20 square cm. 

= 960 square cm. 

(ii) The area of the whole surface of the right pyramid = area of slant surface + area of the base

= (960 + 24 × 24) square cm

= 1536 square cm.

(iii) the volume of the right pyramid

= 1/3 × area of the base × height

= 1/3 × 24 × 24 × 16 cubic cm 

= 3072 cubic cm.

2. The base of a right pyramid 8 m high, is an equilateral triangle of side 12√3 m. Find its volume and the slant surface. Solution:

Let equilateral ∆ WXY be the base and P, the vertex of the right pyramid.

In the plane of the ∆ WXY draw YZ perpendicular to WX and let OZ = 1/3 YZ . Then, O is the centroid of ∆ WXY. Let OP be perpendicular to the plane of ∆ WXY at O; then OP is the height of the pyramid. By question, WX = XY = YW = 8√3 m and OP = 8 m. Since ∆ WXY is equilateral and YZ ┴ WX Hence, Z bisects WX .

Therefore, XZ = 1/2 ∙ WX = 1/2 ∙ 12√3 = 6√3 m. Now, from right - angled ∆ XYZ we get,

YZ² = XY² - XZ²

or, YZ² = (12√3) ² - (6√3)²

or, YZ² = 6² (12 - 3)

or, YZ² = 6² ∙ 9

or, YZ² = 324

Therefore, YZ = 18

Therefore, OZ = 1/3 ∙ 18 = 6. Join PZ . Then, PZ is the slant height of the pyramid. Since OP is perpendicular to the plane of ∆ WXY at O, hence OP ┴ OZ . Therefore, from the right angled ∆ POZ we get,

PZ² = OZ² + OP²

or, PZ ² = 6² + 8²

or, PZ² = 36 + 64

or, PZ² = 100

Therefore, PZ = 10 Therefore, the required slant surface of the right pyramid

= 1/2 × perimetre of the base × slant height

= 1/2 × 3 × 12√3 × PZ

= 1/2 × 36√3 × 10

= 180√3 square meter.

and its volume = 1/3 × area of the base × height

= 1/3 × (√3)/4 (12√3)² × 8

[Since, area of equilateral triangle

= (√3)/4 × (length of a side)² and height = OP = 8]

= 288√3 cubic meter.

●  Mensuration

• Formulas for 3D Shapes
• Volume and Surface Area of the Prism
• Worksheet on Volume and Surface Area of Prism
• Volume and Whole Surface Area of Right Pyramid
• Volume and Whole Surface Area of Tetrahedron
• Volume of a Pyramid
• Volume and Surface Area of a Pyramid
• Worksheet on Volume and Surface Area of a Pyramid
• Worksheet on Volume of a Pyramid

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Surface Area Of A Pyramid

In these lessons, we will learn

• how to find the surface area of any pyramid.
• how to find the surface area of a regular pyramid.
• how to find the surface area of a square pyramid.
• how to find the surface area of a pentagonal and a hexagonal pyramid.
• how to find the surface area of a pyramid when the slant height is not given.

Related Pages Surface Area Formulas Surface Area of Prisms Surface Area of a Sphere More Geometry Lessons

The following diagrams show how to find the surface area of a pyramid. Scroll down the page for more examples and solutions.

A pyramid is a solid with a polygonal base and several triangular lateral faces . The lateral faces meet at a common vertex . The number of lateral faces depends on the number of sides of the base. The height of the pyramid is the perpendicular distance from the base to the vertex.

A regular pyramid has a base that is a regular polygon and a vertex that is above the center of the polygon. A pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle base.

We can find the surface area of any pyramid by adding up the areas of its lateral faces and its base.

Surface area of any pyramid = area of base + area of each of the lateral faces

If the pyramid is a regular pyramid, we can use the formula for the surface area of a regular pyramid. Surface area of regular pyramid = area of base + 1/2 ps where p is the perimeter of the base and s is the slant height.

If the pyramid is a square pyramid, we can use the formula for the surface area of a square pyramid.

Surface area of square pyramid = b 2 + 2 bs where b is the length of the base and s is the slant height.

Worksheets: Calculate the volume of square pyramids Calculate the volume of prisms & pyramids

Since the given pyramid is a square pyramid, we can use any of the above formulas.

Using the formula for the surface area of any pyramid:

Area of base = 6 × 6 = 36 cm 2

Area of the four triangles = 1/2 × 6 × 12 × 4 = 144 cm 2

Total surface area = 36 + 144 = 180 cm 2

Using the formula for a regular pyramid

Surface area of regular pyramid = area of base + 1/2 ps

= 6 × 6 + 1/2 × 6 × 4 × 12 = 180 cm 2

Using the formula for a square pyramid

Surface area of square pyramid = b 2 + 2 bs

= 6 × 6 + 2 × 6 × 12 = 180 cm 2

Surface Area Of Pyramid By Adding Up The Area Of Each Surface

How to find the surface area of a pyramid by adding up the area of each surface? Calculate the surface area of the square based pyramid.

Example: Find the surface area of a square pyramid with s = 40in, h = 39in and n = 44in

Surface Area Of Square Pyramid By Using A Formula

How to find the surface area of a square pyramid using the formula? Surface area = 2 bs + b 2 where b is the length of the base and s is the slant height.

Solve Word Problems With Pyramids

Example: What is the surface area of a square pyramid with a base area of 255 square inches and a height of 7 inches?

The Great Pyramid of Khufu, the largest of the pyramid in Giza, was built approximately 4,500 years ago. Today, the height of the pyramid is about 455 feet, which is about 30 feet shorter than it was originally. If you were to walk completely around the base of the pyramid, you would have gone about 3,024 feet. What is the lateral surface area of the great pyramid today?

Surface Area Of Regular Pyramid By Using A Formula

These videos show how to calculate the surface area of a regular pyramid using the formula: surface area = area of base + 1/2 × perimeter of base × slant height. S = B + 1/2 p l

Surface Area Of Pentagonal And Hexagonal Pyramids

This video provides a specific example of how to find the surface area of a pyramid, given base edge and height. The base is a pentagon. It shows how to find the apothem and slant height.

How To Find The Surface Area Of A Pentagonal Pyramid With The Known Apothem?

Surface Area Of Regular Pyramid When The Slant Height Is Not Given

How to calculate the surface area of a square pyramid when the slant height is not given?

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

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• Surface Area of a Pyramid – Explanation & Examples

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Surface of a pyramid formula

Practice questions, surface area of a pyramid – explanation & examples.

In a pyramid, the lateral faces (which are triangles) meet at a common point known as a vertex. The name of a pyramid is derived from the name of the polygon forming its base. For example, a square pyramid, a rectangular pyramid, a triangular pyramid, a pentagonal pyramid, etc.

The surface area of a pyramid is the sum of the area of the lateral faces. 

This article will discuss how to find the total surface area and lateral surface area of a pyramid .

How to Find the Surface Area of a Pyramid?

To find the surface area of a pyramid, you have to get the area of the base, then add the area of the lateral sides, which is one face times the number of sides.

The general formula for the surface area of any pyramid (regular or irregular) is given as:

Surface area = Base area + Lateral area

Surface area = B + LSA

Where TSA = total surface area

B = base area

LSA = lateral surface area.

For a regular pyramid, the formula is given as:

The total surface area of regular pyramid = B + 1/2 ps

where p = perimeter of the base and s = slant height.

Surface area of a square pyramid

For a square pyramid, the total surface area = b (b + 2s)

Where b = the base length and s = slant height

Surface area of a triangular pyramid

The surface area of a triangular pyramid = ½ b (a + 3s)

Where, a = apothem length of a pyramid

b = base length

s = slant height

Surface area of a pentagonal pyramid

The total surface area of a regular pentagonal pyramid is given by;

Surface area of a pentagonal pyramid = 5⁄2 b (a + s)

Where, a = apothem length of the base

and b = side length of the base, s = slant height of the pyramid

Surface area of the hexagonal pyramid

A hexagonal pyramid is a pyramid with a hexagon as the base.

The total surface area a hexagonal pyramid = 3b (a + s)

Lateral Surface Area of a Pyramid

As stated earlier, the lateral surface area of a pyramid is the area of lateral faces of a pyramid. Since all the lateral faces of a pyramid are triangles, then the pyramid’s lateral surface area is half the product of the perimeter of the pyramid’s base and the slant height.

Lateral surface area (LSA = 1/2 ps)

Where p = perimeter of the base and s = slant height.

Let’s gain an insight into the surface area of a pyramid formula by solving a few example problems.

What is the surface area of a square pyramid whose base length is 4 cm and slant height is 5 cm?

Base length, b = 4 cm

Slant height, s =5 cm

By the formula,

Total surface area of a square pyramid = b (b + 2s)

TSA = 4(4 + 2 x 5)

= 4(4 + 10)

What is the surface area of a square pyramid with a perpendicular height of 8 m and base length of 12 m?

Perpendicular height, h = 8 m

Base length, b =12

To get the slant height s, we apply the Pythagorean Theorem.

s = √ [8 2 + (12/2) 2 ]

s = √ [8 2 + 6 2 ]

s = √ (64 + 36)

Hence, the slant height of the pyramid is 10 m

Now, calculate the surface area of the pyramid.

SA = b (b + 2s)

= 12 (12 + 2 x 10)

= 12(12 + 20)

= 384 m 2 .

Calculate the pyramid’s surface area, whose slant height is 10 ft, and its base is an equilateral triangle of side length, 8 ft.

Base length = 8 ft

Slant height = 10 ft

Apply the Pythagorean theorem to get the apothem length of the pyramid.

a = √ [8 2 – (8/2) 2 ]

= √ (64 – 16)

a = 6.93 ft

Thus, the apothem length of the pyramid is 6.93 ft.

But, the surface area of a triangular pyramid = ½ b (a + 3s)

TSA = ½ x 8(6.93 + 3 x 10)

= 4 (6.93 + 30)

= 4 x 36.93

= 147.72 ft 2

Find the surface area of a pentagonal pyramid whose apothem length is 8 m, base length 6 m, and slant height is 20 m.

Apothem length, a = 8 m

Base length, b = 6 m

Slant height, s = 20 m

TSA = 5/2 x 6(8 + 20)

= 420 m 2 .

Calculate the total surface area and lateral surface area of a hexagonal pyramid with the apothem as 20 m, base length as 18 m, and slant height as 35 m.

apothem, a = 20 m

Base length, b =18 m

Slant height, s = 35 m

The surface area a hexagonal pyramid = 3b (a + s)

= 3 x 18(20 + 35)

= 2,970 m 2 .

The lateral surface area of a pyramid =1/2 ps

Perimeter, p = 6 x 18

LSA = ½ x 108 x 35

= 1,890 m 2

Previous Lesson  |  Main Page | Next Lesson

Surface Area of a Square Pyramid Calculator

Table of contents

Are you in search of a surface area of a square pyramid calculator to estimate all types of areas of the Great pyramid?! If yes, you're in the right place. You can calculate the total surface area, base area, lateral surface area, and face area of any square pyramid using our tool.

We also discuss how to find the surface area of a square pyramid using slant height and base length and how to calculate surface area using slant height and base perimeter . You can also find answers to some interesting numerical problems like the surface area of a pyramid of Giza and the amount of groundsheet required for any tent.

How do I calculate the surface area of a square pyramid?

To find the surface area of a square pyramid:

• Determine how many faces are there on a square pyramid: there are 4 triangular faces. Sum up their areas.
• Find the area of a square base.
• Add up 4 triangular faces area to 1 square base area to find the surface area of a square pyramid.

What is the formula for the surface area of a square pyramid?

The formula to calculate the surface area of a square pyramid is:

or, on simplifying:

• SA is the surface area of the square pyramid;
• a is the length of the base edge; and
• h is the height of the square pyramid.

How do I calculate the base area of a square pyramid?

The base of a square pyramid is a square. Hence, the base area of the square pyramid of base edge a is a 2 .

How to find the lateral area of a square pyramid

First of all, let's explain what the lateral area of a square pyramid is. The lateral surface area or lateral area of a square pyramid is the total area of its 4 triangular faces. We calculate the lateral area as:

💡 The total surface area of a three-dimensional object is the sum of the base area and the lateral surface area !

• SA = BA + LSA

How do I find the face area of a square pyramid?

The face area of a square pyramid is the area of one of its four triangular faces. We calculate the area of a triangular face, the face area (FA), using the formula:

💡 The lateral surface area is the sum of the areas of four triangular faces :

• LSA = 4 × FA

How to find the surface area of a square pyramid using slant height

To find the surface area using the slant height, we use the formula:

• SA = a 2 + 2×a×l

🔎 Proof: The surface area of a square pyramid is the sum of the areas of its square base and four triangular faces:

• SA = BA + (4 × FA)

The area of a triangle is half of the product of its base length ( a ) and height ( l ):

Therefore, the area of four triangular faces or the lateral surface area of the square pyramid is:

• 4 × FA = 2 × a × l

Thus, the lateral surface area (LSA) of the square pyramid of slant height l is

• LSA = 2 × a × l

and the total surface area is

• SA = a 2 + 2 × a × l

How to use the surface area of a square pyramid calculator

❓ The Pyramid of Giza has a height of 480 feet. If the length of each side of the base is approximately 756 feet, what is its total surface area?

Okay, let's see how to use this total surface area of a square pyramid calculator to solve the problem given above:

Identify the measurements :

• Height (h) = 480 ft
• Base edge (a) = 756 ft

Switch the units from cm to ft and cm 2 to ft 2 (or the desired units for the area) using the drop-down list near each variable in the square-based pyramid surface area calculator.

Enter the values for height (480 ft) and base edge (756 ft).

Ta-da! You got the results in no time! Our tool uses the surface area square pyramid formula to find:

• Slant height (l) = 611 ft
• Total surface area (SA) = 1,495,322 ft 2
• Base area (BA) = 571,536 ft 2
• Lateral surface area (LSA) = 923,786 ft 2
• Face area (FA) = 230,947 ft 2

Other related square pyramid calculators

After learning how to calculate the surface area of a square pyramid, you might want to check our other square pyramid calculators:

• Right rectangular pyramid calculator ;
• Square pyramid calculator ;
• Right square pyramid calc ;
• Height of a square pyramid calculator ;
• Rectangular pyramid volume calculator ;
• Surface area of a rectangular pyramid calculator .

How many faces are on a square pyramid?

A square pyramid has 5 faces: 4 equal triangles (side faces) and 1 square (base face).

How do I calculate the surface area of a square pyramid using its base perimeter?

The surface area of a square pyramid is half of the product of its base perimeter and its slant height :

• SA = P × l / 2

where P is the base perimeter and l is the slant height.

How many square meters of groundsheet is required for a tent of base 5m and height 1.8m?

You need (5 m) 2 = 25 m 2 of groundsheet for the tent of base length 5 m irrespective of its height. The amount of groundsheet in square meters is the base area of the tent. The base area of a tent is BA = a 2 , where a is the base length.

Base edge (a)

Slant height (l)

Base perimeter (P)

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Total surface area (SA)

Base area (BA)

Lateral surface area (LSA)

Face area (FA)

• Math Article
• Surface Area Of Pyramid

Surface Area of Pyramid

In Geometry, a pyramid is a three-dimensional shape whose base should be a polygon, and its face should be a triangle. All the triangular side faces meet at a common point called the vertex or apex. The altitude or height of the pyramid is the perpendicular distance from the apex to the centre of the base. Whereas, slant height is a measure of the perpendicular drawn to the base of the side face of a triangle from the apex. In this article, we are going to discuss the surface area of pyramid formulas and examples in detail.

What is the Surface Area of a Pyramid?

The surface area of a pyramid is defined as the sum of the areas of all faces of a pyramid . In other words, the surface area of a pyramid is the total area occupied by the surface of the pyramid. It is generally measured in square units, such as m 2 , cm 2 , in 2 , and so on.

The two types of the surface area of the pyramid are:

• Lateral Surface Area (LSA of Pyramid) = Sum of the area of the side faces of a pyramid.
• Total Surface Area (TSA of Pyramid) = LSA of pyramid + Area of the base.

General Formula for Surface Area of Pyramid

The general formula for the surface area of the pyramid is as follows:

The lateral surface area of the regular pyramid formula is given by:

Similarly, the total surface area of the regular pyramid formula is given by:

Where “l” is the slant height of a pyramid

“P” is the base perimeter of a pyramid

“B” represents the base area of a pyramid

Surface Area of Pyramid Formulas

Generally, if we are asked to find the surface area of the pyramid without any specifications, it represents the total surface area. Now, let us discuss the surface area of pyramid formulas with different bases.

Triangular Pyramid

In the triangular pyramid, the base of the pyramid is a triangle.

Where, b = side length, a = height, s = slant height.

Square Pyramid

In the Square pyramid, the base of the pyramid is a square .

Where, b = side length, s = slant height.

Pentagonal Pyramid

In the pentagonal pyramid, the base of the pyramid is a pentagon.

Where, b = side length, s = slant height, a = apothem length.

Hexagonal Pyramid

In the hexagonal pyramid, the base of the pyramid is a hexagon.

Where, b = side length , s = slant height, a= apothem length.

Surface Area of Pyramid Examples

Go through the solved examples on the surface area of the pyramid:

Determine the surface area of the square pyramid, given that side length = 4 cm and slant height = 8 cm.

Given: Side length, b = 4 cm

Slant height, s = 8 cm

We know that the surface area of square pyramid = 2bs + b 2 Square units

Now, substitute the known values in the formula

S.A = 2(4)(8) + (4) 2

S.A = 64+16 = 80

Hence, the surface area of the square pyramid is 80 cm 2 .

Compute the base area of the pentagonal pyramid, given that side length = 6.4 m and apothem = 16 m.

Given: Side length, b = 6.4 m

Apothem = 16 m

We know that the base of a pentagonal pyramid is a pentagon. Hence, base area = 5⁄2(a × b) square units.

Base area = (5/2)(16×6.4)

Base area = 5(8×6.4) = 5(51.2) = 256

Thus, the base area of a pentagonal pyramid is 256 m 2 .

Practice Questions

Solve the following problems:

• Determine the surface area of the triangular pyramid, given that side length = 2 cm, height = 4 cm and slant height = 5 cm.
• Compute the base area of the square pyramid whose side length is equal to 7 cm.
• Find the surface area of the square pyramid given that side length = 5 cm and slant height = 7 cm.

Stay tuned to BYJU’S – The Learning App and download the app today to learn all Maths-related concepts easily by exploring more videos.

Frequently Asked Questions on Surface Area of Pyramid

What is the surface area of a pyramid.

The surface area of the pyramid is the total area occupied by the surface of the pyramid.

What is the general formula for the lateral surface area of a regular pyramid?

If “P” is the base perimeter and “l” is the slant height, then the lateral surface area of the regular pyramid formula is given by (½) Pl Square units.

What is the general formula for the total surface area of a regular pyramid?

If “P” is the base perimeter, “l” is the slant height, and “B” is the base area, then the total surface area of the regular pyramid formula is given by (½)Pl + B Square units.

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Surface Area of a Pyramid

Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 8 inches and the slant height is 5 inches.

The perimeter of the base is the sum of the sides.

p = 3 ( 8 ) = 24     inches

L . S . A . = 1 2 ( 24 ) ( 5 ) = 60     inches 2

Find the total surface area of a regular pyramid with a square base if each edge of the base measures 16 inches, the slant height of a side is 17 inches and the altitude is 15 inches.

The perimeter of the base is 4 s since it is a square.

p = 4 ( 16 ) = 64     inches

The area of the base is s 2 .

B = 16 2 = 256     inches 2

T . S . A . = 1 2 ( 64 ) ( 17 ) + 256                               = 544 + 256                               = 800     inches 2

There is no formula for a surface area of a non-regular pyramid since slant height is not defined.  To find the area, find the area of each face and the area of the base and add them.

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Surface Area of Pyramid – Formula, Definition With Examples

Updated on January 14, 2024

Welcome to another exciting journey into the world of mathematics with Brighterly ! Today, we will embark on an exploration of the captivating world of pyramids, focusing on their surface area. Pyramids have intrigued mathematicians, architects, and explorers for centuries, from the iconic structures in Egypt to mathematical models in classrooms. Understanding the surface area of a pyramid is more than just a mathematical exercise; it’s an intellectual adventure filled with surprises, shapes, and formulas. Whether you’re a student eager to learn or a curious mind exploring the realms of geometry, our Brighterly guide will take you step by step through everything you need to know. Let’s dive right in!

What Is Surface Area?

A pyramid is a three-dimensional solid object characterized by a polygon base and triangular faces converging at a single point called the apex. The base can be any polygon, such as a triangle, square, or hexagon. Surface Area, on the other hand, refers to the total area covered by all the faces of a three-dimensional object.

Definition of Pyramid

A pyramid is a solid figure with a base that is connected to a single vertex, not in the plane of the base, by line segments from each vertex of the base polygon. It’s a shape that children often see in various cultural depictions, such as the Egyptian Pyramids.

Definition of Surface Area

The surface area of a solid object is the total area that the surface of the object occupies. It encompasses all of the flat surfaces that make up the object, including the base, the faces, and any curved surfaces if present. In simple terms, it’s like wrapping the pyramid in paper and measuring how much paper you need.

Types of Pyramids

Regular pyramids.

Regular Pyramids have a base that is a regular polygon, and all of the faces are congruent, meaning they are the same in shape and size. An example would be a square pyramid, where the base is a square, and all four triangular faces are identical.

Irregular Pyramids

Irregular Pyramids differ from regular pyramids in that the base can be an irregular polygon, and the faces are not necessarily congruent. This adds a unique layer of complexity when calculating the surface area.

Properties of Pyramids

Properties of base.

The base of a pyramid is a polygon that forms its bottom. It can be regular or irregular, which determines the type of pyramid.

Properties of Faces

The faces of a pyramid are the triangular sides that connect the base to the apex. They play a vital role in determining the surface area.

Properties of Apex

The apex is the point where all the faces of a pyramid meet. It’s like the tip of a mountain and a central concept in understanding the pyramid’s structure.

Properties of Surface Area

The properties of surface area of a pyramid include the combined area of its base and faces. Regular pyramids have congruent faces, which simplifies calculations, whereas irregular pyramids may require more detailed measurements.

Importance in Geometry

Understanding pyramids and their surface area is crucial in geometry. It helps in developing spatial understanding, relationships between different shapes, and provides a foundation for further studies in mathematics and architecture.

Difference Between Surface Area and Volume of Pyramid

The surface area is about covering the exterior of the pyramid, while the volume represents the space contained within it. Both have unique formulas and are significant in different contexts, such as design and construction. For example, when building a pyramid-shaped structure, the surface area will determine the amount of material needed to cover the exterior, while the volume may determine the space inside for various applications.

Formula for the Surface Area of Pyramid

Surface area of regular pyramid.

For a regular pyramid, the surface area can be calculated using specific formulas based on the shape of the base and the slant height. For example, in a square pyramid with a base side length of a and slant height l , the surface area would be given by:

Surface Area = a² + 2 a l

Surface Area of Irregular Pyramid

An irregular pyramid may require more complex calculations, as understanding the individual faces’ measurements is essential. The surface area is found by adding the area of the base to the sum of the areas of the triangular faces. For example, if you have an irregular pyramid with a pentagonal base and different measurements for each triangular face, you would need to calculate each face’s area and sum them up.

Practice Problems on Surface Area of Pyramid

Practice makes perfect! Here are some practice problems to reinforce your understanding of pyramids and their surface area.

Find the surface area of a square pyramid with a base side length of 4 cm and slant height of 5 cm.

Calculate the surface area of an irregular pyramid with a triangular base having sides of 3 cm, 4 cm, and 5 cm, and triangular faces with areas of 6 cm², 8 cm², and 10 cm².

And there we have it—a comprehensive and engaging guide to the surface area of pyramids, brought to you by Brighterly. Through the understanding of different types of pyramids, their properties, and the formulas to calculate their surface area, we’ve unraveled an essential aspect of geometry. Pyramids are not just magnificent structures; they’re a testament to human creativity and mathematical brilliance. We hope that this guide will inspire you to look at pyramids with new appreciation and intrigue. Keep exploring, keep learning, and let Brighterly light the path of your mathematical adventure!

Frequently Asked Questions on Surface Area of Pyramid

What is a pyramid.

A pyramid is a three-dimensional solid object with a flat polygon base and triangular faces that converge at a single point called the apex. It has been a subject of fascination in both historical and mathematical contexts, with real-life examples like the Egyptian Pyramids.

How do you calculate the surface area of a pyramid?

The calculation of the surface area depends on whether the pyramid is regular or irregular. For regular pyramids, you can use specific formulas based on the shape of the base and slant height. For irregular pyramids, the process may involve individually measuring each face. Brighterly offers resources and practice problems to make these calculations more accessible.

Why is understanding the surface area of a pyramid important?

The concept of surface area extends beyond mathematics into various applications in real life such as architecture, design, and engineering. Understanding the surface area helps in spatial comprehension, practical problem-solving, and artistic creation. It is a concept taught extensively through Brighterly’s educational resources.

What’s the difference between the surface area and volume of a pyramid?

While surface area refers to the total area covered by the faces of the pyramid, including its base, volume pertains to the space contained within the pyramid. Both have distinct applications and mathematical formulas. Brighterly’s interactive tools can guide learners through these concepts, ensuring a comprehensive understanding.

Can I find examples and practice problems on Brighterly’s website?

Absolutely! Brighterly is dedicated to offering interactive and engaging learning materials. You can find examples, practice problems, and in-depth explanations on the surface area of pyramids in our specially designed resources, tailored to cater to learners of all levels. Explore Brighterly’s Math Section to begin your learning adventure.

• Wikipedia – Pyramid
• Mathworld – Surface Area of a Pyramid
• U.S. Government – Geometry Standards

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.

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How to Find the Surface Area of a Pyramid

Last Updated: November 24, 2023

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 399,033 times.

The surface area of any pyramid can be found by adding the surface area of the base to the surface area of the lateral faces. When working with regular pyramids, you can find the surface area using a formula, as long as you know how to find the area of the base of the pyramid. Since the base can be any polygon, it is helpful to know how to find the area of shapes such as pentagons and hexagons. When working with the common, regular square pyramid, however, calculating the total surface area is a simple calculation, provided you know the slant height of the pyramid and the side length of the square base.

Finding the Surface Area of Any Regular Pyramid

• The basic formula for the surface area of any pyramid, regular or irregular, is Total Surface Area = Base Area + Lateral Area. [2] X Research source
• Don’t confuse “slant height” with “height.” The “slant height” of a regular pyramid is the shortest possible distance from the apex of the pyramid to the edge of the base. [3] X Research source The “height” of a regular pyramid is the distance from the vertex to the center of the base.

Grace Imson, MA

Our Expert Agrees: The surface area of a pyramid is equal to the sum of the areas of all of the faces. First, you have to get the area of the base, then add the area of the lateral sides, which is one face times the number of sides.

Finding the Surface Area of a Square Pyramid

• Don’t confuse “slant height” with “height.” The “slant height” is the diagonal distance from the apex of the pyramid to the edge of the base. [4] X Research source The “height” is the perpendicular distance from the vertex to the base.

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About This Article

To find the surface area of a pyramid, start by multiplying the perimeter of the pyramid by its slant height. Then, divide that number by 2. Finally, add the number you get to the area of the pyramid's base to find the surface area. To learn how to find the surface area of a square pyramid, scroll down! Did this summary help you? Yes No

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Surface Area of a Square Pyramid

In this section, we will learn about the surface area of a square pyramid. A pyramid is a 3-D object whose all side faces are congruent triangles and whereas its base can be any polygon. One side of each of these triangles coincides with one side of the base polygon. A square pyramid is a pyramid whose base is a square. The pyramids are named according to the shape of their bases. Just like other three-dimensional shapes, a square pyramid also has two types of areas.

• Total Surface Area (TSA)
• Lateral Surface Area (LSA)

Let us learn about the surface area of a square pyramid along with the formula and a few solved examples here. You can find a few practice questions in the end.

What is the Surface Area of a Square Pyramid?

The word "surface" means " the exterior or outside part of an object or body". So, the total surface area of a square pyramid is the sum of the areas of its lateral faces and its base. We know that a square pyramid has:

• a base which is a square.
• 4 side faces, each of which is a triangle .

All these triangles are isosceles and congruent, each of which has a side that coincides with a side of the base ( square ).

So, the surface area of a square pyramid is the sum of the areas of four of its triangular side faces and the base area which is square.

Formula of Surface Area of a Square Pyramid

Let us consider a square pyramid whose base's length (square's side length) is 'a' and the height of each side face (triangle) is 'l' (this is also known as the slant height). i.e., the base and height of each of the 4 triangular faces are 'a' and 'l' respectively. So the base area of the pyramid which is a square is a × a = a 2 and the area of each such triangular face is 1/2 × a × l. So the sum of areas of all 4 triangular faces is 4 ( ½ al) = 2 al. Let us now understand the formulas to calculate the lateral and total surface area of a square pyramid using height and slant height.

Total Surface Area of Square Pyramid Using Slant Height

The total surface area of a square pyramid is the total area covered by the four triangular faces and a square base. The total surface area of a square pyramid using slant height can be given by the formula, Surface area of a square pyramid = a 2 + 2al where,

• a = base length of square pyramid
• l = slant height or height of each side face

Total Surface Area of a Square Pyramid Using Height

Let us assume that the height of the pyramid (altitude) be 'h'. Then by applying Pythagoras theorem (you can refer to the below figure),

\(l = \sqrt{\dfrac{a^{2}}{4}+h^{2}}\)

Substituting this in the above formula,

The surface area of a square pyramid = a 2 + 2al = a 2 + 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)

Note: \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\) can be simplified as \(\dfrac 1 2 \sqrt{a^2+4h^2}\). Thus, the formula of surface area of a square pyramid can be written as a 2 + 2a \(\left(\dfrac 1 2 \sqrt{a^2+4h^2}\right)\) = a 2 + a\( \sqrt{a^2+4h^2}\).

Lateral Surface Area of a Square Pyramid

The lateral surface area of a square pyramid is the area covered by the four triangular faces. The lateral surface area of a square pyramid using slant height can be given by the formula, Lateral surface area of a square pyramid = 2 al or, Lateral surface area of a square pyramid = 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\) where,

• h = height of square pyramid

How to Calculate Surface Area of Square Pyramid?

The surface area of a square pyramid can be calculated by representing the 3D figure into a 2D net. After expanding the 3D figure into a 2D net we will get one square and four triangles.

• To find the area of the square base: a 2 , 'a' is the base length.
• To find the area of the four triangular faces: The area of the four triangular side faces can be given as: 2al, 'l' is the slant height. If slant height is not given, we can calculate it using height, 'h' and base length as, \(l = \sqrt{\dfrac{a^{2}}{4}+h^{2}}\)
• Add all the areas together for the total surface area of a square pyramid, while the area of 4 triangular faces gives the lateral area of the square pyramid.
• Thus, the surface area of a square pyramid is a 2 + 2al and lateral surface area as 2al in squared units.

Now, that we have seen the formula and method to calculate the surface area of a square pyramid, let us have a look at a few solved examples to understand it better.

Examples on Surface Area of a Square Pyramid

Example 1: Find the surface area of a square pyramid of slant height 15 units and base length 12 units.

The base length of the square pyramid is, a = 12 units.

Its slant height is, l = 15 units.

The surface area = a 2 +2al = 12 2 +2 (12) (15) = 504 units 2

Answer: The surface area of the given square pyramid is 504 units 2 .

Example 2: The height of a square pyramid is 25 units and the base area of a square pyramid is 256 square units. Find its surface area.

Let the side of the base (square) be 'a' units.

Then it is given that a 2 = 256 ⇒ a = 16 units.

The height of the given square pyramid is h = 25 units.

Surface area of square pyramid = a 2 + 2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)

= 16 2 +2 (16) \(\sqrt{\dfrac{16^{2}}{4}+25^{2}}\) ≈ 1095.96 square units.

Answer: The surface area of the given square pyramid = 1095.96 square units.

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Practice Questions on Surface Area of a Square Pyramid

Faqs on surface area of a square pyramid, what is the surface area of the square pyramid.

The surface area of a square pyramid is the sum of the areas of all its 4 triangular side faces with the base area of the square pyramid. If a, h, and l are the base length, the height of the pyramid, and slant height respectively, then the surface area of the square pyramid = a 2 + 2al (or) a 2 +2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

How Do You Find the Lateral Area of a Square Pyramid?

To find the lateral area of a square pyramid, find the area of one side face (triangle) and multiply it by 4. If a and l are the base length and the slant height of a square pyramid, then the lateral area of the square pyramid = 4 (½ × a × l) = 2al.

If h is the height of the pyramid, then the lateral area = 2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

What Is the Area of One of the Triangular Faces of a Square Pyramid?

If a and l are the base length and the slant height of a square pyramid, then the area of one of the 4 triangular side faces is, ½ × a × l.

How Do You Find the Lateral Area and Surface Area of a Square Pyramid?

The lateral area of a square pyramid is the sum of the areas of the side faces only, whereas the surface area is the lateral area + area of the base. The lateral area of a square pyramid = 2al (or) 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

To get the total surface area, we need to add the area of the base (which is a 2 ) to each of these formulas. The total surface area of a square pyramid = a 2 + 2al (or) a 2 + 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\). where,

• a = Length of the base (square)
• l = Slant height
• h = Height of the pyramid

How To Calculate Surface Area of a Square Pyramid Without Slant Height?

We know, slant height of a square pyramid is given in terms of height and base length by the formula, \(l = \sqrt{\dfrac{a^{2}}{4}+h^{2}}\). We can calculate the slant height from the given height and base length and apply the formula for surface area of square pyramid as, LSA of pyramid = 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\) TSA of pyramid = a 2 + 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\) where,

What Is the Base Area of a Square Pyramid?

The base of a square pyramid is square-shaped. Thus, the base area of square pyramid can be calculated using the formula, Base Area of Pyramid = a 2 , where, a is the length of the base of square pyramid.

How Many Bases Does a Square Pyramid Have?

A square pyramid is a pyramid with a square-shaped base. A square pyramid thus has only one base.

Which Two Shapes Make up a Square Pyramid?

The base of a square pyramid is a square and its side faces are triangles. So the two shapes that make up a square pyramid are square and triangle.

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Pyramid problems.

Surface area and volume of pyramid problems along with with detailed solutions are presented.

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Surface Area Worksheets

Surface area worksheets comprise an enormous collection of exercises on different solid figures. The large chunk of exercises is categorized based on a step-by-step approach involving counting unit squares to determine the SA, finding the surface area of nets, and then computing the surface area of geometrical shapes like cubes, cones, cylinders, rectangular prisms, L-shaped prisms, spheres, hemispheres, pyramids, and composite figures, catering to the needs of students in 5th grade through high school. Revisiting surface area is easy with our mixed 3D shapes revision pdfs. Count on our free printable SA worksheets for a great start!

List of Surface Area Worksheets

• Surface Area of 3D Figures Using Nets
• Surface Area by Counting Squares
• Surface Area of Cubes
• Surface Area of Rectangular Prisms
• Surface Area of L-Shaped Prisms
• Surface Area of Triangular Prisms
• Surface Area of Cylinders
• Surface Area of Prisms and Cylinders
• Surface Area of Cones
• Surface Area of Spheres and Hemispheres
• Surface Area of Pyramids
• Surface Area of Mixed Solid Shapes
• Surface Area of Composite Figures

Explore the Surface Area Worksheets in Detail

Surface Area of 3D Figures Using Nets Worksheets

Finding the surface area of 3D figures using nets worksheets assist students in visualizing the surface of solid shapes whose nets are sketched on grids. Add on to their practice in determining the SA of the nets of 3D shapes like cylinders, cones, and pyramids, and in drawing the nets of solid figures too.

Surface Area by Counting Squares Worksheets

Start off with counting unit squares on an isometric paper, follow up by drawing the correct number of squares, and then find the SA of rectangular prisms by counting the squares scaled to varied units. These interesting exercises make the surface area by counting squares pdfs a compulsive print for your grade 5 and grade 6 students.

Surface Area of Cubes Worksheets

Page through these surface area of a cube exercises to practice computing the total area occupied by the cubes with edge length offered as integers, decimals and fractions. Included here are pdfs to find the missing edge length using the SA and more.

Surface Area of Rectangular Prisms Worksheets

Surface area of rectangular prisms handouts are a sure-fire hit in every grade 6, grade 7, and grade 8 geometry curriculum. Find the SA using the height, width, and length, and extend your practice to finding the missing dimensions as well.

Surface Area of L-Shaped Prisms Worksheets

Lay a strong foundation in decomposing shapes with these printable surface area of L-shaped rectangular prism worksheets. Solve for surface area of independent 3D shapes, add them, and subtract the area of the face that connects the rectangular blocks.

Surface Area of Triangular Prisms Worksheets

Expedite practice calculating the surface area of triangular prisms with these worksheet pdfs, rendering the attributes of the triangular bases and rectangular side faces of the figure, in integers and decimals.

Surface Area of Cylinders Worksheets

The surface area of a cylinder pdfs feature solid shapes, each made up of a curved surface with two circular bases. Recall the formula for SA of a cylinder, plug in the integer, decimal, or fractional radius measures in the formula 2πrh + 2πr 2 and compute.

Surface Area of Prisms and Cylinders Worksheets

A mix of rectangular prisms, triangular prisms, and cylinders is what these printable worksheets have in store for your middle and high school learners. Consider memorizing the SA formulas, apply the one relevant to the solid shape, substitute the dimensions and solve.

Surface Area of Cones Worksheets

Give your 6th grade, 7th grade, and 8th grade students an edge over their peers with these surface area of cones exercises. Supplying the values of the dimensions in the formula and calculating the surface area of cones is all that is expected of learners.

Surface Area of Spheres and Hemispheres Worksheets

Build fluency and competence with this collection of surface area of spheres and hemispheres worksheets. Get students to find the TSA of the hemispheres and CSA of the spheres by substituting the dimensions in relevant formulas.

Surface Area of Pyramids Worksheets

Handle these printable surface area of pyramid practice sheets with great dexterity. Learn the know-how of finding the surface area of pyramids applying relevant formulas and substituting the dimensions accordingly.

Surface Area of Mixed Solid Shapes Worksheets

How well do your 8th grade and high school students remember the SA formulas of solid shapes like cubes, cones, cylinders, spheres, hemispheres, prisms and pyramids? Check for yourself with these surface area of solid shapes revision pdfs.

Surface Area of Composite Figures Worksheets

Tee up to decompose each combined shape, find the sum of the SA of individual 3D shapes, subtract the area of the common parts, and determine the SA like a pro with this set of printable surface area of composite shapes worksheets.

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Surface area

Here you will learn about surface area, including what it is and how to calculate it for prisms and pyramids.

Students will first learn about surface area as part of geometry in 6 th grade.

What is surface area?

The surface area is the total area of all of the faces of a three-dimensional shape. This includes prisms and pyramids. The surface area is always recorded in square units.

Prisms are 3D shapes that have a polygonal base and rectangular faces. A rectangular prism has 6 rectangular faces, including 4 rectangular lateral faces and 2 rectangular bases.

For example,

Calculate the area of each face and then add them together for the surface area of the rectangular prism.

The surface area of the prism is the sum of the areas. Add each area twice, since each rectangle appears twice in the prism:

8+8+12+12+6+6=52 \, f t^2

You can also find the surface area by multiplying each area by 2 and then adding.

(2 \times 8)+(2 \times 12)+(2 \times 6)=52 \, f t^2

Step-by-step guide: Surface area of rectangular prism

[FREE] Surface Area Worksheet (Grade 6 to 8)

Use this worksheet to check your grade 6 to 8 students’ understanding of surface area. 15 questions with answers to identify areas of strength and support!

Another type of prism is a triangular prism.

A triangular prism is made up of 5 faces, including 2 triangular bases and 3 rectangular lateral faces.

Calculate the area of each face and then add them together for the surface area of the triangular prism.

The surface area of the prism is the sum of the areas. Add the area of the triangular base twice (or you can multiply it by 2 ), since it appears twice in the prism:

37.2+60+38.4+21+21=177.6 \mathrm{~mm}^2

Step-by-step guide: Surface area of triangular prism

Step-by-step guide: Surface area of a prism

Pyramids are another type of 3D shape. A pyramid is made up of a polygonal base and triangular lateral sides.

All lateral faces (sides) of this square pyramid are congruent.

To calculate the surface area of a pyramid , calculate the area of each face of the pyramid and then add the areas together.

\text {Area of the base }=2.5 \times 2.5=6.25 \mathrm{~cm}^2

\text {Area of a triangular face }=\cfrac{1}{2} \times 2.5 \times 4=5 \mathrm{~cm}^2

Add the area of the base and the 4 congruent triangular faces:

\text {Surface area }=6.25+5+5+5+5=6.25+(4 \times 5)=26.25 \mathrm{~cm}^2

The total surface area can also be written in one equation:

​​\begin{aligned} \text {Surface area of pyramid } & =\text {Area of base }+ \text {Areas of triangular faces } \\\\ & =2.5^2+4 \times\left(\cfrac{1}{2} \, \times 2.5 \times 4\right) \\\\ & =6.25+20 \\\\ & =26.25 \mathrm{~cm}^2 \end{aligned}

Step-by-step guide: Surface area of a pyramid

Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Geometry (6.G.A.4) Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

How to calculate the surface area of a prism

In order to calculate the surface area:

Calculate the area of each face.

Add the area of each face together.

Include the units.

Surface area examples

Example 1: surface area of a rectangular prism.

Calculate the surface area of the rectangular prism.

A rectangular prism has 6 faces, with 3 pairs of identical faces.

2 Add the area of each face together.

Total surface area: 14+14+21+21+6+6=82

Since opposite rectangles are always congruent, you can also use multiplication to solve:

Total surface area: 14 \times 2+21 \times 2+6 \times 2=82

3 Include the units.

The measurements on this prism are in m , so the total surface area of the prism is 82 \mathrm{~m}^2.

Example 2: surface area of a triangular prism with an equilateral triangle – using a net

Calculate the surface area of the triangular prism. The base of the prism is an equilateral triangle with a perimeter of 16.5 \, ft.

First, use the perimeter of the base to find the length of each side. Since an equilateral triangle has all equal sides, s , the perimeter is s+s+s=16.5.

s=5.5 \, ft

You can unfold the triangular prism, and use the net to find the area of each face:

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 5.5 \, ft side of the triangle would combine to form an edge with each corresponding rectangle – making their lengths equal.

The area of each triangular base:

\cfrac{1}{2} \times 4.8 \times 5.5=13.2

The area of each rectangular lateral face:

10 \times 5.5=55

If you have trouble keeping track of all the calculations, use a net:

The area of the base is always equal to the area of the opposite base, in this case the triangles.

Notice, since the triangle is equilateral, all the rectangular faces are equal as well.

Total surface area: 13.2+13.2+55+55+55=191.4

The measurements on this prism are in ft , so the total surface area of the prism is 191.4 \mathrm{~ft}^2.

Example 3: surface area of a square-based pyramid in cm

All the lateral faces of the pyramid are congruent. Calculate the surface area.

The base is a square with the area 6\times{6}=36\text{~cm}^2.

All four triangular faces are identical, so calculate the area of one triangle, and then multiply the area by 4 .

\begin{aligned} A&= \cfrac{1}{2} \, \times{b}\times{h}\\\\ &=\cfrac{1}{2} \, \times{10}\times{6}\\\\ &=30 \end{aligned}

30\times{4}=120

Add the area of the base and the area of the four triangles:

SA=36+120=156

The side lengths are measured in centimeters, so the area is measured in square centimeters.

SA=156\text{~cm}^{2}

Example 4: surface area of a rectangular prism – using a net

Calculate the lateral surface area of the rectangular prism. The base of the prism is a square and one side of the base measures 3 \, \cfrac{2}{3} inches.

You can unfold the rectangular prism, and use the net to find the area of each face:

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 3 \cfrac{2}{3} \mathrm{~ft} side of the square would combine to form an edge with each corresponding rectangle – making their lengths equal.

\begin{aligned} & 9 \cfrac{4}{5} \, \times 3 \, \cfrac{2}{3} \\\\ &= \cfrac{49}{5} \, \times \cfrac{11}{3} \\\\ &= \cfrac{539}{15} \\\\ &= 35 \, \cfrac{14}{15} \end{aligned}

Remember, you are only finding the area of the lateral faces, so you do not need to calculate the area of the bases.

Notice, since the square has all equal sides, all the rectangular faces are equal as well.

Total lateral surface area:

\begin{aligned} & 35 \, \cfrac{14}{15}+35 \, \cfrac{14}{15}+35 \, \cfrac{14}{15}+35 \, \cfrac{14}{15} \\\\ & =140 \, \cfrac{56}{15} \\\\ & =143 \, \cfrac{11}{15} \end{aligned}

Since all the lateral faces are congruent, you can also use multiplication to solve:

\begin{aligned} & 4 \times 35 \, \cfrac{14}{15} \\\\ & =\cfrac{4}{1} \, \times \, \cfrac{539}{15} \\\\ & =\cfrac{2,156}{15} \\\\ & =143 \, \cfrac{11}{15} \end{aligned}

The measurements on this prism are in inches, so the total lateral surface area of the prism is 143 \, \cfrac{11}{15} \text {~inches }^2.

Example 5: surface area of a parallelogram prism with different units

Calculate the surface area of the parallelogram prism.

A parallelogram prism has 6 faces and, like a rectangular prism, it has 3 pairs of identical faces. The base is a parallelogram and all of the lateral faces are rectangular.

In this example, some of the measurements are in cm and some are in m . You must convert the units so that they are the same. Convert all the units to meters (m)\text{: } 40 {~cm}=0.4 {~m} and 50 {~cm}=0.5 {~m}.

Total surface area: 0.48+0.48+1.8+1.8+0.75+0.75=6.06

The measurements that we have used are in m so the surface area of the prism is 6.06 \mathrm{~m}^2.

Example 6: surface area of a square pyramid – word problem

Mara is making a square pyramid out of cardboard. She cut out 4 acute triangles that have a base of 5 inches and a height of 7.4 inches. How much cardboard will she need to complete the entire square pyramid?

The lateral faces are all congruent, acute triangles.

\begin{aligned} \text {Area of triangle } & =\cfrac{1}{2} \, \times 5 \times 7.4 \\\\ & =18.5 \end{aligned}

Since it is a square pyramid, the base is a square. Each side of the square shares an edge with the base of the triangle, so each side of the square is 5 .

\begin{aligned} \text { Area of square } & =5 \times 5 \\\\ & =25 \end{aligned}

There is one square base and 4 congruent lateral triangular faces.

Total surface area: 25+18.5 \times 4=25+74=99

The measurements on this prism are in inches, so the total surface area of the prism is 99 \text {~inches}^2.

Teaching tips for the surface area of a prism

• Make sure that students have had time to work with physical 3D models and nets before doing activities that involve finding the surface area of pyramids and prisms.
• Choose worksheets that offer a variety of question types – a mixture of showing the full pyramid or prism versus showing the net, a mixture of solving for the missing surface area versus a missing dimension, and one that includes some word problems.

Easy mistakes to make

• Confusing lateral area with total surface area Lateral area is the area of each of the sides, and total surface area is the area of the bases plus the area of the sides. When asked to find the lateral area, be sure to only add up the area of the sides – which are always rectangles in right prisms (the types of prisms shown on this page). Note: In oblique prisms the lateral faces are parallelograms.

Practice surface area of a prism questions

1) The pyramid is composed of four congruent equilateral triangles. Find the surface area of the pyramid.

\begin{aligned} \text {Surface area of pyramid }&= \text { Area of base and faces} \\ & \quad \text{ (4 congruent triangles) } \\\\ & =4 \times\left(\cfrac{1}{2} \, \times 3 \times 2.6\right) \\\\ & =4 \times 3.9 \\\\ & =15.6 \mathrm{~ft}^2 \end{aligned}

2) Calculate the surface area of the triangular prism:

You can unfold the triangular prism, and use the net to find the area of each face.

Remember that the edges in a prism always fold up together to form the prism – making their lengths equal.

\cfrac{1}{2} \times 5 \times 4.3=10.75

7 \times 5=35

Total surface area: 10.75+10.75+35+35+35=126.5 \mathrm{~ft}^2

3) Calculate the surface area of the rectangular prism:

You can unfold the rectangular prism, and use the net to find the area of each face.

The area of each rectangular base:

\begin{aligned} & 1 \cfrac{2}{5} \, \times 8 \\\\ &= \cfrac{7}{5} \, \times \cfrac{8}{1} \\\\ &= \cfrac{56}{5} \\\\ &= 11 \, \cfrac{1}{5} \end{aligned}

\begin{aligned} & 4 \, \cfrac{2}{3} \, \times 8 \\\\ &= \cfrac{14}{3} \, \times \cfrac{8}{1} \\\\ &= \cfrac{112}{3} \\\\ &= 37 \, \cfrac{1}{3} \end{aligned}

\begin{aligned} & 4 \, \cfrac{2}{3} \, \times 1 \cfrac{2}{5} \\\\ &= \cfrac{14}{3} \, \times \cfrac{7}{5} \\\\ &= \cfrac{98}{15} \\\\ &= 6 \cfrac{8}{15} \end{aligned}

Total surface area:

\begin{aligned} & 6 \, \cfrac{8}{15} \, +6 \, \cfrac{8}{15} \, +11 \, \cfrac{1}{5} \, +11 \, \cfrac{1}{5} \, +37 \, \cfrac{1}{3} \, +37 \, \cfrac{1}{3} \\\\ & =6 \, \cfrac{8}{15} \, +6 \, \cfrac{8}{15} \, +11 \, \cfrac{5}{15} , +11 \, \cfrac{5}{15} \, +37 \, \cfrac{3}{15} \, +37 \, \cfrac{3}{15} \\\\ & =108 \, \cfrac{32}{15} \\\\ & =110 \, \cfrac{2}{15} \mathrm{~m}^2 \end{aligned}

4) Here is a net of a square pyramid. Calculate the surface area.

\begin{aligned} \text {Area of triangle } & =\cfrac{1}{2} \, \times 6.5 \times 3.8 \\\\ & =12.35 \end{aligned}

Since it is a square pyramid, the base is a square.

\begin{aligned} \text {Area of square } & =6.5 \times 6.5 \\\\ & =42.25 \end{aligned}

Total surface area = 12.35+12.35+12.35+12.35+42.25=91.65 \mathrm{~m}^2

5) Calculate the surface area of the prism.

The congruent bases (front and back faces) are composed of a rectangle and a right triangle.

Total surface area = 87.5+87.5+330+220+154+189.2=1,068 .2 \text { units}^2 

6) Malika was painting the hexagonal prism below. It took 140.8 \text { inches}^2 to cover the entire shape. If the area of the base is \text {10.4 inches}^2 and each side of the hexagon is 2 \text { inches} , what is the height of the prism?

You can unfold the hexagonal prism, and use the net to find the area of each face:

Total area of the bases: 10.4+10.4=20.8

Subtract the area of the bases from the total amount of paint Malika used, to see how much was used on the lateral faces:

140.8-20.8=120

The total area of the faces left is 120 \text { inches}^2. 

Since the 6  faces are congruent, the total for each face can be found by dividing by 6\text{:} 

120 \div 6=20 

Labeling the missing length as x , means the area of each face can be written as 2 \times x or 2 x .

Since each face has an area of 20 \text{ inches}^2 , the missing height can be found with the equation: 2 x=20.

Since 2 \times 10=20 , the missing height is 10 inches.

Surface area FAQs

A cuboid is a prism with a rectangular base and rectangular lateral sides. It is also known as a rectangular prism.

Some shapes do have a general formula that you can use. For example, the surface area of a rectangular prism uses the formula 2 \: (l b+b h+l h) . There are other formulas, but for all prisms, the general formula is \text {area of } 2 \text { bases }+ \text {area of all lateral faces} .

Since all the faces have the same area, find the area of the square base and multiply it by 6 . Step-by-step guide : Surface area of a cube

The surface area of a cylinder is the area of a circle (the two congruent bases) plus the the curved surface area (2 \pi r h). . This will give you the surface area of the cylinder. Step-by-step guide: Surface area of a cylinder

To find the curved surface area, square the radius of the sphere and multiply it by 4 \pi . This will give you the surface area of the sphere. Step-by-step guide : Surface area of a sphere

The next lessons are

• Pythagorean theorem
• Trigonometry
• Circle math
• Surface area of a cone
• Surface area of a hemisphere

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