Interior and Exterior Angles of Regular Polygons Word Problems
The angles that lie inside a shape, generally a polygon, are said to be interior angles .
An exterior angle of a polygon is the angle that is formed between any side of the polygon and a line extended from the next side. Every polygon has interior and exterior angles. The exterior is the term opposite to the interior which means outside. Therefore exterior angles can be found outside the polygon. The sum of the exterior angles of any polygon is equal to 360°. Any flat shape or figure is said to have interior or exterior angles only if it is a closed shape.
To find the sum of all interior angles in a regular polygon: The sum of the interior angles of a polygon can be found by taking the number of sides (n) and subtracting 2. Then, multiply that number by 180.
Sum of interior angles = (n – 2) ∙ 180°
To find the measure of each interior angle in a regular polygon: 1. To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below. Sum of interior angles = (n – 2) ∙ 180°
2. Next, divide the sum of interior angles by the total number of angles the regular polygon has.
To find the measure of each exterior angle in a regular polygon: The measure of one of the exterior angles of a regular polygon can be found by dividing 360 degrees by the number of angles (n).
Bees build honeycombs with hexagonal cells. What is the measure of each interior angle of the cell?
To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below.
A hexagon has 6 sides, so: (6 – 2) ∙ 180 = 4 ∙ 180 = 720°
Since this is a regular hexagon, all of the angles are equal, so divide the sum of the interior angles by 6.
Practice Interior and Exterior Angles of Regular Polygons Word Problems
Practice Problem 1
Practice Problem 2
Practice Problem 3
Polygon – A closed figure formed by three or more segments called sides.
Interior angle – An angle of a polygon formed by two of its side and is inside the polygon.
Exterior angle – An angle formed by one side and the extension of the adjacent side. It is outside the polygon.
Pre-requisite Skills Classify Angles Drawing Angles Estimating Angles Angle Relationships Classify Triangles Angles Finding Angle Measures Complementary and Supplementary Angles Angles in Triangles
Related Skill Geometric Proof
PROBLEMS ON INTERIOR AND EXTERIOR ANGLES OF TRIANGLE
Problem 1 :
In triangle PQR, the measure of ∠ P is 36. The measure of ∠ Q is five times the measure of ∠ R. Find ∠ Q and ∠ R.
∠ P = 36
∠ Q = 5 ∠ R
In triangle PQR, the sum of interior angles is 180° .
∠ P + ∠ Q + ∠ R = 180°
36° + 5 ∠ R + ∠ R = 180°
6R + 36° = 180°
Subtract 36° from both sides.
6 ∠ R = 144°
Divide both sides by 6.
∠ R = 24 °
∠ Q = 5 ∠ R
∠ Q = 5(24° )
∠ Q = 120 °
Problem 2 :
The exterior angle of a triangle is 120° . Find the value of x if the opposite non-adjacent interior angles are (4x + 40)° and 60°.
Using Exterior Angle Theorem,
(4x + 40)° + 60 ° = 120°
4x + 40 + 60 = 120
4x + 100 = 120
Subtract 100 from both sides.
Divide both sides by 4.
Problem 3 :
Find the values of x , y and z .
x° + z° = 56° ----(1)
y° and 56° are linear pair.
y° + 56° = 180°
Subtract 56 from both sides.
x° and 144° are linear pair.
x° + 144° = 180°
Subtract 144 from both sides.
x = 36
Substitute x = 36 in (1).
36 + z = 56
Subtract 36 from both sides.
z = 20
Problem 4 :
y° = 26° + 26°
y = 52
By Angle Sum Property of Triangle,
x° + y° + 64° = 180°
X + 52 + 64 = 180
x + 116 = 180
Subtract 116 from both sides.
x = 64
Problem 5 :
x° + 33° = z°
x + 33 = z ----(1)
z° and 133° are linear pair.
z + 133 = 180
Subtract 133 from both sides.
z = 47
Substitute z = 47 in (1).
x + 33 = 47
Subtract 33 from both sides.
x = 14
y° + z° +114° = 180°
y + z + 114 = 180
Subtract 114 from both sides.
y + z = 66
Substitute z = 47.
y + 47 = 66
y = 19
Problem 6 :
Find the values of i and n .
Vertically opposite angles are equal.
n° = 114° + 33°
n = 147
94° + i° + 33° = 180°
i + 127 = 180
Subtract 127 from both sides.
i = 53
Kindly mail your feedback to [email protected]
We always appreciate your feedback.
© All rights reserved. onlinemath4all.com
- Sat Math Practice
- SAT Math Worksheets
- PEMDAS Rule
- BODMAS rule
- GEMDAS Order of Operations
- Math Calculators
- Transformations of Functions
- Order of rotational symmetry
- Lines of symmetry
- Compound Angles
- Quantitative Aptitude Tricks
- Trigonometric ratio table
- Word Problems
- Times Table Shortcuts
- 10th CBSE solution
- PSAT Math Preparation
- Privacy Policy
- Laws of Exponents
Recent Articles
Simplifying Algebraic Expressions with Fractional Coefficients
May 17, 24 08:12 AM
The Mean Value Theorem Worksheet
May 14, 24 08:53 AM
Mean Value Theorem
May 14, 24 02:48 AM
- school Campus Bookshelves
- menu_book Bookshelves
- perm_media Learning Objects
- login Login
- how_to_reg Request Instructor Account
- hub Instructor Commons
Margin Size
- Download Page (PDF)
- Download Full Book (PDF)
- Periodic Table
- Physics Constants
- Scientific Calculator
- Reference & Cite
- Tools expand_more
- Readability
selected template will load here
This action is not available.
4.18: Exterior Angles and Theorems
- Last updated
- Save as PDF
- Page ID 4815
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
Exterior angles equal the sum of the remote interiors.
Exterior Angles
An Exterior Angle is the angle formed by one side of a polygon and the extension of the adjacent side.
In all polygons, there are two sets of exterior angles, one that goes around clockwise and the other goes around counterclockwise.
Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to \(180^{\circ}\).
\(m\angle 1+m\angle 2=180^{\circ} \)
There are two important theorems to know involving exterior angles: the Exterior Angle Sum Theorem and the Exterior Angle Theorem.
The Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to \(360^{\circ}\).
\(m\angle 1+m\angle 2+m\angle 3=360^{\circ}\)
\(m\angle 4+m\angle 5+m\angle 6=360^{\circ}\).
The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of its remote interior angles . ( Remote Interior Angles are the two interior angles in a triangle that are not adjacent to the indicated exterior angle.)
\(m\angle A+m\angle B=m\angle ACD\)
What if you knew that two of the exterior angles of a triangle measured \(130^{\circ}\)? How could you find the measure of the third exterior angle?
Example \(\PageIndex{1}\)
Two interior angles of a triangle are \(40^{\circ}\) and \(73^{\circ}\). What are the measures of the three exterior angles of the triangle?
Remember that every interior angle forms a linear pair (adds up to \(180^{\circ}\)) with an exterior angle. So, since one of the interior angles is \(40^{\circ}\) that means that one of the exterior angles is \(140^{\circ}\) (because \(40+140=180\)). Similarly, since another one of the interior angles is \(73^{\circ}\), one of the exterior angles must be \(107^{\circ}\). The third interior angle is not given to us, but we could figure it out using the Triangle Sum Theorem . We can also use the Exterior Angle Sum Theorem. If two of the exterior angles are \(140^{\circ}\) and \(107^{\circ}\), then the third Exterior Angle must be \(113^{\circ}\) since \(140+107+113=360\).
So, the measures of the three exterior angles are 140, 107 and 113.
Example \(\PageIndex{2}\)
Find the value of \(x\) and the measure of each angle.
Set up an equation using the Exterior Angle Theorem.
\(\begin{align*} \underbrace{(4x+2)^{\circ}+(2x−9)^{\circ}}_\text{remote interior angles}&=\underbrace{(5x+13)^{\circ}}_\text{exterior angle} \\ (6x−7)^{\circ}&=(5x+13)^{\circ} \\ x&=20 \end{align*}\)
Substitute in 20 for \(x\) to find each angle.
\([4(20)+2]^{\circ}=82^{\circ}[2(20)−9]^{\circ}=31^{\circ} \qquad Exterior \:angle:\: [5(20)+13]^{\circ}=113^{\circ}\)
Example \(\PageIndex{3}\)
Find the measure of \(\angle RQS\).
Notice that \(112^{\circ}\) is an exterior angle of \(\Delta RQS\) and is supplementary to \(\angle RQS\).
Set up an equation to solve for the missing angle.
\(\begin{align*}112^{\circ}+m\angle RQS &=180^{\circ} \\ m\angle RQS&=68^{\circ}\end{align*}\)
Example \(\PageIndex{4}\)
Find the measures of the numbered interior and exterior angles in the triangle.
We know that \(m\angle 1+92^{\circ}=180^{\circ}\) because they form a linear pair. So, m\angle 1=88^{\circ}\).
Similarly, \(m\angle 2+123^{\circ}=180^{\circ}\) because they form a linear pair. So, m\angle 2=57^{\circ}\).
We also know that the three interior angles must add up to 180^{\circ}\) by the Triangle Sum Theorem.
\(\begin{align*} m\angle 1+m\angle 2+m\angle 3&=180^{\circ} \qquad by\: the \:Triangle \:Sum \:Theorem. \\ 88^{\circ}+57^{\circ}+m\angle 3&=180 \\ m\angle 3&=35^{\circ}\end{align*}\)
Lastly, \(m\angle 3+m\angle 4=180^{\circ} \qquad because\: they\: form \:a \:linear \:pair.\)
\(\begin{align*} 35^{\circ}+m\angle 4&=180^{\circ} \\ m\angle 4&=145^{\circ}\end{align*}\)
Example \(\PageIndex{5}\)
What is the value of \(p\) in the triangle below?
First, we need to find the missing exterior angle, which we will call \(x\). Set up an equation using the Exterior Angle Sum Theorem.
\(\begin{align*} 130^{\circ}+110^{\circ}+x&=360^{\circ} \\ x&=360^{\circ}−130^{\circ}−110^{\circ} \\ x&=120^{\circ}\end{align*} \)
\(x\) and \(p\) add up to \(180^{\circ}\) because they are a linear pair.
\(\begin{align*} x+p&=180^{\circ} \\ 120^{\circ}+p&=180^{\circ} \\ p&=60^{\circ}\end{align*}\)
Determine \(m\angle 1\).
Use the following picture for the next three problems:
- What is \(m\angle 1+m\angle 2+m\angle 3\)?
- What is \(m\angle 4+m\angle 5+m\angle 6\)?
- What is \(m\angle 7+m\angle 8+m\angle 9\)?
Solve for \(x\).
Additional Resources
Interactive Element
Video: Exterior Angles Theorems Examples - Basic
Activities: Exterior Angles Theorems Discussion Questions
Study Aids: Triangle Relationships Study Guide
Practice: Exterior Angles and Theorems
Real World: Exterior Angles Theorem
Angles of Polygons
In these lessons, we will learn
- how to calculate the sum of interior angles of a polygon using the sum of angles in a triangle
- the formula for the sum of interior angles in a polygon
- how to solve problems using the sum of interior angles
- the formula for the sum of exterior angles in a polygon
- how to solve problems using the sum of exterior angles.
All the polygons in these lessons are assumed to be convex polygons .
Related Pages Polygons Quadrilaterals Cyclic Quadrilaterals More Geometry Lessons
The following diagrams give the formulas for the sum of the interior angles of a polygon and the sum of exterior angles of a polygon. Scroll down the page if you need more examples and explanation.
Sum Of Interior Angles Of A Polygon
We first start with a triangle (which is a polygon with the fewest number of sides). We know that
The sum of interior angles in a triangle is 180°.
This is also called the Triangle Sum Theorem. Click here if you need a proof of the Triangle Sum Theorem.
Next, we can figure out the sum of interior angles of any polygon by dividing the polygon into triangles. We can separate a polygon into triangles by drawing all the diagonals that can be drawn from one single vertex.
In the quadrilateral shown below, we can draw only one diagonal from vertex A to vertex B. So, a quadrilateral can be separated into two triangles.
The sum of angles in a triangle is 180°. Since a quadrilateral is made up of two triangles the sum of its angles would be 180° × 2 = 360°
The sum of interior angles in a quadrilateral is 360º
A pentagon (five-sided polygon) can be divided into three triangles. The sum of its angles will be 180° × 3 = 540°
The sum of interior angles in a pentagon is 540°.
A hexagon (six-sided polygon) can be divided into four triangles. The sum of its angles will be 180° × 4 = 720°
The sum of interior angles in a hexagon is 720°.
Formula For The Sum Of Interior Angles
We can see from the above examples that the number of triangles in a polygon is always two less than the number of sides of the polygon. We can then generalize the results for a n-sided polygon to get a formula to find the sum of the interior angles of any polygon.
The following diagram shows the formula for the sum of interior angles of an n-sided polygon and the size of an interior angle of a n-sided regular polygon. Scroll down the page for more examples and solutions on the interior angles of a polygon.
Example: Find the sum of the interior angles of a heptagon (7-sided)
Solution: Step 1: Write down the formula (n - 2) × 180°
Step 2: Plug in the values to get (7 - 2) × 180° = 5 × 180° = 900°
Answer: The sum of the interior angles of a heptagon (7-sided) is 900°.
Example: Find the interior angle of a regular octagon.
Answer: Each interior angle of an octagon (8-sided) is 135°.
Worksheet using the Formula for the Sum of Interior Angles
How to find the sum of the interior angles of any polygon using triangles and then derive the generalized formula?
Problems Using The Sum Of Interior Angles
How to find a missing angle using the sum of interior angles of a polygon?
How to use the sum of interior angles to write an equation and solve for the unknown? Write an equation and solve for the unknown. Substitute your answer into each expression to determine the measure of the angle. Give reasons for your answers.
Formula For The Sum Of Exterior Angles
The sum of exterior angles of any polygon is 360°.
The exterior angle of a regular n-sided polygon is 360°/n
Worksheet using the formula for the sum of exterior angles
Worksheet using the formula for the sum of interior and exterior angles
How to find the sum of the exterior angles and interior angles of a polygon? Every convex polygon has interior and exterior angles. The interior angles are inside the polygon formed by the sides. The exterior angles form a linear pair with the interior angles.
Example: Determine the measure of each exterior and interior angle of a regular polygon.
Problems using the sum of exterior angles
The following video shows a problem involving the sum of exterior angles of a polygon.
Example: A regular polygon has an exterior angle that measures 40°. How many sides does the polygon have?
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.
One to one maths interventions built for KS4 success
Weekly online one to one GCSE maths revision lessons now available
In order to access this I need to be confident with:
This topic is relevant for:
Interior Angles Of A Polygon
Here we will learn about interior angles in polygons including how to calculate the sum of interior angles for a polygon, single interior angles and use this knowledge to solve problems.
There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
What are interior angles?
Interior angles are the angles inside a shape. They are the angles within a polygon made by two sides:
Interior and exterior angles form a straight line – they add to 180° :
We can calculate the sum of the interior angles of a polygon by splitting it into triangles and multiplying the number of triangles by 180° .
The number of triangles a polygon can be split into is always 2 less than the number of sides.
A heptagon has 7 sides.
7 – 2 = 5 , so we can split the heptagon into 5 triangles:
The general formula is:
Sum of Interior Angles = (n – 2) × 180° ‘ n ’ is the number of sides the polygon has
Step by step guide: Angles in polygons
What are interior angles of polygons?
- Polygon : A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines.
- Regular & irregular polygons : A regular polygon is where all angles are equal size and all sides are equal length E.g. a square An irregular polygon is where all angles are not equal size and/or all sides are not equal length E.g. a trapezium.
How to solve problems involving interior angles
In order to solve problems involving interior angles:
- Identify the number of sides in any polygon/s given in the question. Note whether the shape is regular or irregular.
Find the sum of interior angles for any polygon/s given.
- Identify what the question is asking.
Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of exterior angles \, \textbf{=} \, \bf{360^{\circ}} if required
How to solve problems involving interior angles of a polygon.
Interior and exterior angles worksheet (includes interior angles of a polygon)
Get your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Related lessons on angles in polygons
Interior angles of a polygon is part of our series of lessons to support revision on angles in polygons . You may find it helpful to start with the main angles in polygons lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
- Angles in polygons
- Exterior angles of a polygon
- Angles in a triangle
- Angles in a quadrilateral
- Angles in a pentagon
- Angles in a hexagon
Interior angles examples
Example 1: finding a single interior angle of a regular polygon.
Find the size of each interior angle for a regular decagon.
- Identify the number of sides in any polygon/s given in the question. Note whether this are regular or irregular shapes.
10 sides – regular shape.
2 Find the sum of interior angles for any polygon/s given.
Sum of interior angles = (n – 2) × 180°
As a decagon has 10 sides:
n=10 , so we can substitute n=10 into the formula.
Sum of interior angles of a decagon = (10 – 2) × 180°
Sum of interior angles of a decagon = 8 × 180°
Sum of interior angles of a decagon = 1440°
3 Identify what the question is asking you to find.
The question is asking for ‘each interior angle’. This means the size of one interior angle.
4 Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of exterior angles \, \textbf{=} \, \bf{360^{\circ}} if required
We know the sum of the interior angles for this polygon is 1440° .
We know, as it is a regular polygon, that all the angles are of equal size.
Therefore we can find the size of each interior angle by dividing the sum of interior angles by the number of angles in the polygon:
The size of each interior angle is 144° .
Step-by-step guide: Substitution
Example 2: finding a single interior angle of an irregular polygon
The diagram shows a polygon. Find the size of angle x .
Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes.
6 sides – irregular hexagon
Sum of interior angles = (n – 2) × 180°
Sum of interior angles for a hexagon = (6 – 2) × 180°
Sum of interior angles for a hexagon = 720°
Identify what the question is asking you to find.
Finding the missing angle labelled as x .
Note that we know the values of all the other angles.
The size of angle is 119° .
Example 3: finding the number of sides given the interior angle of a regular polygon
Each of the interior angles of a regular polygon is 140° . How many sides does the polygon have?
Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes.
Unknown number of sides – regular shape
We need to find the number of sides.
We know a single angle of this regular polygon is 140° .
Therefore all the angles are 140° .
We can write the sum of the interior angles as 140 multiplied by the number of sides or 140n .
The polygon has 9 sides.
Note: We can also solve this problem by calculating an exterior angle.
Example 4: multiple shapes
Shown below are three congruent regular pentagons. Find angle y .
Each polygon has 5 sides (pentagon) and is regular.
As each polygon shown is a regular pentagon they all have equal sums of their interior angles:
Sum of interior angles for a pentagon = (5 – 2) × 180°
Sum of interior angles for each pentagon = 540°
Find the missing angle y shown on the diagram.
We know that angles around a point add to 360° , so if we add the three interior angles shown and y together we will get 360° .
Each interior angle shown is 540 ÷ 5 = 108°
We can now calculate y by forming an equation:
Angle y is equal to 36° .
Example 5: problem solving to find the number of sides
Shown below are sections of three identical regular polygons where AB, BC and CA are all sides of the polygons.
ABC is an equilateral triangle formed by placing the three larger polygons together.
Calculate the number of sides each regular polygon has.
Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes .
Shown is an equilateral triangle (regular shape) made up of the adjacent sides AB, BC and CA .
We need to calculate the number of sides of the larger polygons.
An equilateral triangles has the sum of interior angles of 180° .
We do not know the number of sides of the polygons so their sum of interior angles can be represented by (n – 2) × 180° .
The number of sides of the regular polygons where we are only shown one side.
Looking at point A we can see there are three angles around a point. One of the angles is within the equilateral triangle, so it must be 60° , and the other two angles are from the polygons we are attempting to find.
We will call these angles x :
We know that angles around a point add to 360° .
This means that each interior angle of the regular polygon is 150° .
So the sum of interior angles is equal to 150 × n or 150n :
150n = (n – 2) × 180
We can now solve for n :
The polygon has 12 sides, so each polygon shown in the diagram has 1 2 sides.
Example 6: problem solving to find the number of angles
Shown is a regular pentagon. Find y .
5 sides – regular
Sum of interior angles for a decagon = (5 – 2) × 180°
Sum of interior angles for a decagon = 540°
Find angle y which is within one of the interior angles.
As the polygon is regular you can find the size of one interior angle by:
540° ÷ 5 = 108
As the polygon is regular AC = AB
Therefore ABC is an isosceles triangle where angles ACB and ABC are equal to one another and are therefore both y .
We know that the interior angles of a triangle add to 180° .
Common misconceptions
- Miscounting the number of sides
- Misidentifying if a polygon is regular or irregular
- Dividing the sum of interior angles by the number of triangles created. You should divide by the number of sides to find the size of one interior angle (for regular polygons only)
- Incorrectly assuming all the angles are the same size
- Misidentifying which angle the questions is asking you to calculate
Practice interior angles of a polygon questions
1. Find the sum of interior angles for a polygon with 13 sides
Sum of Interior Angles = (n-2)\times180
In this case n=13 , so the calculation becomes 11 \times 180 .
2. Find the size of one interior angle for a regular quadrilateral
The sum of interior angles in a quadrilateral is 360^{\circ} . For a regular shape all the angles are the same size, so we divide 360 by 4 to arrive at the answer.
3. Find the size of one interior angle for a regular nonagon
The sum of interior angles in a nonagon is 1260^{\circ} . For a regular shape all the angles are the same size, so we divide 1260 by 9 to arrive at the answer.
4. Each of the interior angles of a regular polygon is 165^{\circ} . How many sides does the polygon have?
With this in mind, we have 165n=(n-2) \times 180
Which simplifies to 15n = 360
5. Each of the interior angles of a regular polygon is 160^{\circ} . How many sides does the polygon have?
With this in mind, we have 160n=(n-2)\times180
Which simplifies to 20n = 360
6. Four interior angles in a pentagon are each 115^{\circ} . Find the size of the other angle.
By using the formula,
We know that a pentagon has interior angles that add up to 540^{\circ} .
540 – (4 \times 155) = 80
Interior angles of a polygon gcse questions.
1. Work out the size of the angle labeled x .
(6-2) \times 180 = 720
80 + 55 + 280 + 25 + 162 = 602
720-602=118
2. The diagram below shows a regular decagon.
(a) Work out the size of angle a .
(b) Work out the size of angle b .
(10-2) \times 180 = 1440
1440 \div 10 = 144
144\times 2 = 288
360 – 288 = 72
72 \div 2=36
3. A regular polygon’s interior and exterior angles are in the ratio 9 : 1 . How many sides does the polygon have?
180^{\circ} in ratio 9 : 1
180 \div 10=18, 18 \times 9=162, 18 \times 1= 18
One interior angle = 162^{\circ}
\begin{aligned} 162n&=(n-2) \times 180 \\ 162n&=180n-360 \end{aligned}
Learning checklist
You have now learned how to:
- Use conventional terms for geometry e.g. interior angle
- Knowing names and properties of polygons
- Calculate the sum of interior angles for a regular polygon
- Derive and use the sum of angles in a triangle to deduce and use the angle sum in any polygon, and to derive properties of regular polygons
- Calculate the size of the interior angle of a regular polygon
The next lessons are
- Angle rules
- Angles in parallel lines
- How to calculate volume
Still stuck?
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths tuition programme.
Privacy Overview
Angles in Polygons Practice Questions
Click here for questions, click here for answers.
missing, polygon, angle
GCSE Revision Cards
5-a-day Workbooks
Primary Study Cards
Privacy Policy
Terms and Conditions
Corbettmaths © 2012 – 2024
- Parallelogram
- Quadrilateral
- Parallelepiped
- Tetrahedron
- Dodecahedron
- Fraction Calculator
- Mixed Fraction Calculator
- Greatest Common Factor Calulator
- Decimal to Fraction Calculator
- Whole Numbers
- Rational Numbers
- Place Value
- Irrational Numbers
- Natural Numbers
- Binary Operation
- Numerator and Denominator
- Order of Operations (PEMDAS)
- Scientific Notation
Table of Contents
Last modified on August 3rd, 2023
#ezw_tco-2 .ez-toc-title{ font-size: 120%; ; ; } #ezw_tco-2 .ez-toc-widget-container ul.ez-toc-list li.active{ background-color: #ededed; } chapter outline
Interior and exterior angles.
All angles are classified based on whether they are found inside or outside of any geometric shape into interior and exterior angles.
What are Interior Angles
Angles that are found inside or within any geometric shape are called interior angles. They are also sometimes called internal angles.
A triangle has three interior angles. Similarly a quadrilateral such as a square, rectangle, parallelogram, kite, or a trapezoid has four interior angles. Again polygons such as pentagon, hexagon, heptagon, octagon, nonagon, and decagon have five, six, seven, eight, nine, and ten interior angles.
Shown below are the interior angles of some common regular polygons.
Sum of Interior Angles
To obtain the sum of interior angles we simply add the measures of all the angles found within the shape. For example, the three angles of a triangle add up to 180°. Similarly quadrilaterals add up to 360°.
To make the process less tedious, the sum of interior angles in all regular polygons is calculated using the formula given below:
Sum of interior angles = ( n -2) x 180°, here n = here n = total number of sides
Let us take an example to understand the concept,
For an equilateral triangle, n = 3
Sum of interior angles of an equilateral triangle = (n-2) x 180°
= (3-2) x 180°
Find the sum of the interior angles of a square.
As we know, Sum of interior angles = (n-2) x 180°, here n = 4 = (4-2) x 180° = 360°
One Interior Angle
To find the measure of a single interior angle of a regular polygon, we simply divide the sum of the interior angles value with the total number of sides. For an irregular polygon, the unknown angle can be determined when measure of all other angles and their sum are known.
The formula for determining one interior angle in a regular polygon is given below:
One interior angle = ( n -2) x 180°/ n , here n = total number of sides
Let us take an example to understand the concept better,
One interior angle = ( n -2) x 180°/ n , here n = 3
= (3-2) x 180°/3
Let us take some more examples to understand the concepts better.
Find the measure of one interior angle of a regular dodecagon.
As we know, One interior angle = ( n -2) x 180°/ n , here n = 12 = (12 – 2) x 180°/12 = 150°
Find the measure of the unknown interior angle in an irregular hexagon with angles 130°, 90°, 140°, 150°, and 90°.
As we know, Sum of interior angles = (n-2) x 180°, here n = 6 = (6-2) x 180° = 720° Let the unknown angle be x° Now, 130° + 90° + 140° + 150° + 90° + x = 720° x = 720° – 600° x = 120°
What are Exterior Angles
Angles that are found outside or external to any geometric shape are called exterior angles. They are also sometimes known as external angles or turning angle. An exterior angle is made by extending one of the lines of the shape beyond the point of intersection.
(show only the exterior angles and not interior angles)
Show all the shapes given in the link and mark all their exterior angles. Write the number of interior angles with number alongside the figure.
Sum of Exterior Angles
Since an exterior angle is formed by extending a side, the sum of the interior and the exterior angle on the same vertex of any polygon is 180°. The formula to determine the sum of exterior angles is derived below:
Now, for any polygon with n sides,
Sum of exterior angles + Sum of interior angles = n x 180°
Sum of exterior angles = n x 180° – Sum of all interior angles… (1)
Putting the formula for sum of all interior angles in (1) we get,
Sum of exterior angles = n x 180° – (n-2) x 180°
= n x 180° – (n x 180° + 2 x 180°)
= 180°n – 180°n + 360°
Sum of the exterior angles of any polygon is 360°.
One Exterior Angle
To find the measure of a single exterior angle, we simply divide the measure of sum of the exterior angles with the total number of sides. The formula to determine one exterior angle is given below:
One exterior angle = 360°/ n , here n = total number of sides
Also the value of an exterior angle can be obtained by subtracting the interior angle from 180°.
One exterior angle = 180° – Adjacent interior angle
Let us take some examples to understand the concept better,
Find the measure of the exterior angle of a decagon when its corresponding interior angle is 144°.
As we know, Exterior angle = 180° – Adjacent interior angle, here interior angle = 144° = 180° – 144° =36°
Find the measure of each exterior angle of a regular dodecagon.
As we know, One exterior angle = 360°/ n , here n = 12 = 360°/12 = 30°
Each exterior angle of a regular polygon is 18 ° . Find the number of sides of the polygon
As we know, One exterior angle = 360°/ n , here n = total number of sides Thus, Total number of sides = 360°/ one exterior angle, here one exterior angle = 18 = 360°/18 = 20°
One thought on “ Interior and Exterior Angles ”
Thankyou so much , this information is very helpful to me.
Leave a comment Cancel reply
Your email address will not be published. Required fields are marked *
Save my name, email, and website in this browser for the next time I comment.
- Privacy Policy
- Trigonometry
Join Our Newsletter
© 2024 Mathmonks.com . All rights reserved. Reproduction in whole or in part without permission is prohibited.
Remote, Exterior and Interior Angles of A Triangle
Formula for remote, exterior and interior angles, what are the remote and interior angles.
An exterior angle of a triangle , or any polygon , is formed by extending one of the sides.
In a triangle, each exterior angle has two remote interior angles . The remote interior angles are just the two angles that are inside the triangle and opposite from the exterior angle.
As the picture above shows, the formula for remote and interior angles states that the measure of a an exterior angle $$ \angle A $$ equals the sum of the remote interior angles.
To rephrase it, the angle 'outside the triangle' (exterior angle A) equals D + C (the sum of the remote interior angles).
--> exterior angle $$ \angle 1 = 110° $$ --> $$ \angle 2 $$ ? -->