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Volume Problem Solving

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To solve problems on this page, you should be familiar with the following: Volume - Cuboid Volume - Sphere Volume - Cylinder Volume - Pyramid

This wiki includes several problems motivated to enhance problem-solving skills. Before getting started, recall the following formulas:

• Volume of sphere with radius \(r:\) \( \frac43 \pi r^3 \)
• Volume of cube with side length \(L:\) \( L^3 \)
• Volume of cone with radius \(r\) and height \(h:\) \( \frac13\pi r^2h \)
• Volume of cylinder with radius \(r\) and height \(h:\) \( \pi r^2h\)
• Volume of a cuboid with length \(l\), breadth \(b\), and height \(h:\) \(lbh\)

Volume Problem Solving - Basic

Volume - problem solving - intermediate, volume problem solving - advanced.

This section revolves around the basic understanding of volume and using the formulas for finding the volume. A couple of examples are followed by several problems to try.

Find the volume of a cube of side length \(10\text{ cm}\). \[\begin{align} (\text {Volume of a cube}) & = {(\text {Side length}})^{3}\\ & = {10}^{3}\\ & = 1000 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

Find the volume of a cuboid of length \(10\text{ cm}\), breadth \(8\text{ cm}\). and height \(6\text{ cm}\). \[\begin{align} (\text {Area of a cuboid}) & = l × b × h\\ & = 10 × 8 × 6\\ & = 480 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

I made a large ice cream cone of a composite shape of a cone and a hemisphere. If the height of the cone is 10 and the diameter of both the cone and the hemisphere is 6, what is the volume of this ice cream cone? The volume of the composite figure is the sum of the volume of the cone and the volume of the hemisphere. Recall the formulas for the following two volumes: \( V_{\text{cone}} = \frac13 \pi r^2 h\) and \( V_{\text{sphere}} =\frac43 \pi r^3 \). Since the volume of a hemisphere is half the volume of a a sphere of the same radius, the total volume for this problem is \[\frac13 \pi r^2 h + \frac12 \cdot \frac43 \pi r^3. \] With height \(h =10\), and diameter \(d = 6\) or radius \(r = \frac d2 = 3 \), the total volume is \(48\pi. \ _\square \)

Find the volume of a cone having slant height \(17\text{ cm}\) and radius of the base \(15\text{ cm}\). Let \(h\) denote the height of the cone, then \[\begin{align} (\text{slant height}) &=\sqrt {h^2 + r^2}\\ 17&= \sqrt {h^2 + 15^2}\\ 289&= h^2 + 225\\ h^2&=64\\ h& = 8. \end{align}\] Since the formula for the volume of a cone is \(\dfrac {1}{3} ×\pi ×r^2×h\), the volume of the cone is \[ \frac {1}{3}×3.14× 225 × 8= 1884 ~\big(\text{cm}^{2}\big). \ _\square\]

Find the volume of the following figure which depicts a cone and an hemisphere, up to \(2\) decimal places. In this figure, the shape of the base of the cone is circular and the whole flat part of the hemisphere exactly coincides with the base of the cone (in other words, the base of the cone and the flat part of the hemisphere are the same). Use \(\pi=\frac{22}{7}.\) \[\begin{align} (\text{Volume of cone}) & = \dfrac {1}{3} \pi r^2 h\\ & = \dfrac {1 × 22 × 36 × 8}{3 × 7}\\ & = \dfrac {6336}{21} = 301.71 \\\\ (\text{Volume of hemisphere}) & = \dfrac {2}{3} \pi r^3\\ & = \dfrac {2 × 22 × 216}{3 × 7}\\ & = \dfrac {9504}{21} = 452.57 \\\\ (\text{Total volume of figure}) & = (301.71 + 452.57) \\ & = 754.28.\ _\square \end{align} \]

Try the following problems.

Find the volume (in \(\text{cm}^3\)) of a cube of side length \(5\text{ cm} \).

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

Bob has a pipe with a diameter of \(\frac { 6 }{ \sqrt { \pi } }\text{ cm} \) and a length of \(3\text{ m}\). How much water could be in this pipe at any one time, in \(\text{cm}^3?\)

What is the volume of the octahedron inside this \(8 \text{ in}^3\) cube?

A sector with radius \(10\text{ cm}\) and central angle \(45^\circ\) is to be made into a right circular cone. Find the volume of the cone.

\[\] Details and Assumptions:

• The arc length of the sector is equal to the circumference of the base of the cone.

Three identical tanks are shown above. The spheres in a given tank are the same size and packed wall-to-wall. If the tanks are filled to the top with water, then which tank would contain the most water?

A chocolate shop sells its products in 3 different shapes: a cylindrical bar, a spherical ball, and a cone. These 3 shapes are of the same height and radius, as shown in the picture. Which of these choices would give you the most chocolate?

\[\text{ I. A full cylindrical bar } \hspace{.4cm} \text{ or } \hspace{.45cm} \text{ II. A ball plus a cone }\]

How many cubes measuring 2 units on one side must be added to a cube measuring 8 units on one side to form a cube measuring 12 units on one side?

This section involves a deeper understanding of volume and the formulas to find the volume. Here are a couple of worked out examples followed by several "Try It Yourself" problems:

\(12\) spheres of the same size are made from melting a solid cylinder of \(16\text{ cm}\) diameter and \(2\text{ cm}\) height. Find the diameter of each sphere. Use \(\pi=\frac{22}{7}.\) The volume of the cylinder is \[\pi× r^2 × h = \frac {22×8^2×2}{7}= \frac {2816}{7}.\] Let the radius of each sphere be \(r\text{ cm}.\) Then the volume of each sphere in \(\text{cm}^3\) is \[\dfrac {4×22×r^3}{3×7} = \dfrac{88×r^3}{21}.\] Since the number of spheres is \(\frac {\text{Volume of cylinder}}{\text {Volume of 1 sphere}},\) \[\begin{align} 12 &= \dfrac{2816×21}{7×88×r^3}\\ &= \dfrac {96}{r^3}\\ r^3 &= \dfrac {96}{12}\\ &= 8\\ \Rightarrow r &= 2. \end{align}\] Therefore, the diameter of each sphere is \[2\times r = 2\times 2 = 4 ~(\text{cm}). \ _\square\]

Find the volume of a hemispherical shell whose outer radius is \(7\text{ cm}\) and inner radius is \(3\text{ cm}\), up to \(2\) decimal places. We have \[\begin{align} (\text {Volume of inner hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × R^3\\ & = \dfrac {1 × 4 × 22 × 27}{2 × 3 × 7}\\ & = \dfrac {396}{7}\\ & = 56.57 ~\big(\text{cm}^{3}\big) \\\\ (\text {Volume of outer hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × r^3\\ & = \dfrac {1 × 4 × 22 × 343}{2 × 3 × 7}\\ & = \dfrac {2156}{7}\\ & = 718.66 ~\big(\text{cm}^{3}\big) \\\\ (\text{Volume of hemispherical shell}) & = (\text{V. of outer hemisphere}) - (\text{V. of inner hemisphere})\\ & = 718.66 - 56.57 \\ & = 662.09 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

A student did an experiment using a cone, a sphere, and a cylinder each having the same radius and height. He started with the cylinder full of liquid and then poured it into the cone until the cone was full. Then, he began pouring the remaining liquid from the cylinder into the sphere. What was the result which he observed?

There are two identical right circular cones each of height \(2\text{ cm}.\) They are placed vertically, with their apex pointing downwards, and one cone is vertically above the other. At the start, the upper cone is full of water and the lower cone is empty.

Water drips down through a hole in the apex of the upper cone into the lower cone. When the height of water in the upper cone is \(1\text{ cm},\) what is the height of water in the lower cone (in \(\text{cm}\))?

On each face of a cuboid, the sum of its perimeter and its area is written. The numbers recorded this way are 16, 24, and 31, each written on a pair of opposite sides of the cuboid. The volume of the cuboid lies between \(\text{__________}.\)

A cube rests inside a sphere such that each vertex touches the sphere. The radius of the sphere is \(6 \text{ cm}.\) Determine the volume of the cube.

If the volume of the cube can be expressed in the form of \(a\sqrt{3} \text{ cm}^{3}\), find the value of \(a\).

A sphere has volume \(x \text{ m}^3 \) and surface area \(x \text{ m}^2 \). Keeping its diameter as body diagonal, a cube is made which has volume \(a \text{ m}^3 \) and surface area \(b \text{ m}^2 \). What is the ratio \(a:b?\)

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction \( \frac{m}{n} \), for relatively prime integers \(m\) and \(n\). Compute \(m+n\).

The square-based pyramid A is inscribed within a cube while the tetrahedral pyramid B has its sides equal to the square's diagonal (red) as shown.

Which pyramid has more volume?

Please remember this section contains highly advanced problems of volume. Here it goes:

Cube \(ABCDEFGH\), labeled as shown above, has edge length \(1\) and is cut by a plane passing through vertex \(D\) and the midpoints \(M\) and \(N\) of \(\overline{AB}\) and \(\overline{CG}\) respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

If the American NFL regulation football

has a tip-to-tip length of \(11\) inches and a largest round circumference of \(22\) in the middle, then the volume of the American football is \(\text{____________}.\)

Note: The American NFL regulation football is not an ellipsoid. The long cross-section consists of two circular arcs meeting at the tips. Don't use the volume formula for an ellipsoid.

Answer is in cubic inches.

Consider a solid formed by the intersection of three orthogonal cylinders, each of diameter \( D = 10 \).

What is the volume of this solid?

Consider a tetrahedron with side lengths \(2, 3, 3, 4, 5, 5\). The largest possible volume of this tetrahedron has the form \( \frac {a \sqrt{b}}{c}\), where \(b\) is an integer that's not divisible by the square of any prime, \(a\) and \(c\) are positive, coprime integers. What is the value of \(a+b+c\)?

Let there be a solid characterized by the equation \[{ \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.\]

Calculate the volume of this solid if \(a = b =2\) and \(c = 3\).

• Surface Area

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• Math Article

Volume Of A Cube

Volume of a cube is the total cubic units occupied by it, in a three-dimensional space. A cube is a 3d-shape, that has six faces, twelve edges and eight vertices. Hence, the volume of a cube is the space enclosed by its six faces. Unlike, 2d shapes , it has additional dimensions apart from length and width, which is called height or thickness.  Therefore, the volume of cube is equal to product of its length, width and height. It is measured in cubic units. The more the value of its dimensions the more is the volume of cube.

In this article, we will discuss the definition and formula of volume of a cube along with examples.

What is the Volume of a Cube?

Definition: The volume of a cube defines the number of cubic units, occupied by the cube completely. The unit of volume of the cube is in cubic units such as centimeters 3 , meters 3 , inches 3 , feet 3 , etc.

A cube is a solid three-dimensional figure, which has 6 square faces or sides. The volume of a cube will be the total space occupied by it. Since all the faces of the cube are square in the shape, hence, the length of edges will also be equal. Therefore, the length, width and height of the cube are of equal length.

If length, width and height of cube is equal to ‘a’, then;

Volume of cube = a  × a × a

If we know the edge length i.e. “a”, then we can find the volume of the cube. Let us learn how to find the volume of any cubical structure.

Volume of a Cube Formula

We can easily find the volume of the cube (V), by knowing the length of its edges. Suppose, the length of the edges of the cube is ‘a’. Then the V will be the product of length, height, and width. So, the volume of the cube formula is:

Volume of a Cube = Length × Width × Height

Volume = a × a × a

Volume = a 3

Where ‘a’ is the length of the side of the cube or edges.

Derivation for Volume of a Cube

The volume of an object is defined as the amount of space a solid occupies. We know that a cube is a 3-dimensional object whose all the sides i.e. length, breadth, and height are equal. Now for a cube, the volume derivation will be as follows:

• Consider a square sheet of paper.
• Now, the area that the square sheet will take up will be its surface area i.e. its length multiplied by its breadth.
• As the square will have equal length and breadth, the surface area will be “a 2 “.
• Now, a cube is made by stacking multiple square sheets on top of each other so that the height becomes “a” units. This gives the height or thickness of the cube as “a”.
• Now, it can be concluded that the overall area covered by the cube will be the area of the base multiplied by the height.
• So,  Volume of Cube = a 2  × a = a 3

How to Find Volume of a Cube?

There are two methods by which we can find the volume of a cube.

• Using Edge-length
• Using Diagonal

Both the methods are formula-based and easy to understand.

Volume of Cube Given the Edge-Length

By the formula of volume of a cube, we know that;

Volume = (Edge of the cube) 3

Therefore, if we know the edge length of the cube, we can easily find its volume.

Example: If edge-length is 3 cm, the volume of the cube;

Volume = 3 3 = 3 x 3 x 3 = 27 cu.cm.

Volume of a Cube Given the Diagonal

The volume of a cube whose diagonal (d), is given is given by:

Hence, by measuring the length of the diagonal of cube, we can evaluate the volume of cube.

Example: If length of diagonal of cube is 3 inches, then the volume will be:

Volume = √3 × d 3 /9 = √3 × 3 3 /9 = 3√3 cu.in.

Check:  Diagonal of a Cube Formula

Video Lesson on Volume of Cube

For more information on volumes of cubes and cuboid, watch the below video:.

Surface Area of a Cube

The surface area of the cube is the total area occupied by its outer surface. We can find the surface area of a cube, which is equal to the number of square units that cover the surface of the cube, completely. The general formula of surface area for a cube of side ‘a’, is given by;

Surface Area of Cube = 6a 2

Related Articles

• Cube and Cuboid
• Surface Area Of Cube
• Volume of a Cone
• Volume Of Cuboid
• Volume of a Cylinder
• Volume Of Sphere
• Volume of a Pyramid
• Volume of a Prism

Solved Examples on Volume of Cube

Question 1: Find the volume of the cube, having the sides of length 7 cm.

Given, the length of the sides of the cube is 7 cm.

We know, Volume of a cube = (length of sides of the cube) 3

Therefore, Volume, V = (7 cm) 3

V = 343 cm 3

Question 2: Find the length of the edges of the cube, if its volume is equal to 125 cm 3 .

Given, Volume of the cube = 125 cm 3 .

Let the length of the edges is ‘a’.

We know, by the formula,

The volume of a cube = (length of edges of the cube) 3

Substituting the value, we get,

Or a = 3 √125

Or a = 5 cm

Therefore, the length of the cube is 5 cm.

Question.3 : What is the volume of cube, whose length of diagonal is 27 cm.

Solution : By the formula of volume of cube given the diagonal, we have;

Volume = √3 × d 3 /9

d = 27 cm (given)

Volume of cube = √3 × 27 3 /9

= √3 × (27 × 27 × 27)/9

= √3 × (27 × 27 × 3)

= 3788 (Approx)

Practice Questions

• What is the volume of cube if length of diagonal is 15 cm?

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Frequently Asked Questions on Volume of a Cube

What is meant by the volume of a cube, what is the formula of volume of cube, what is the unit of volume of cube, what is the relationship of the volume of the cube to its edge length, what is the volume of a cube when diagonals are given.

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Volume Of A Cube

Here we will learn about the volume of a cube, including how to calculate the volume of a cube and how to find missing lengths of a cube given its volume.

There are also volume of a cube worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the volume of a cube?

The volume of a cube is the amount of space there is within a cube. 

A cube is a three-dimensional shape with 6 square faces.

To find the volume of a cube, with side length a, we can use the volume of a cube formula,

Volume is measured in cubic units for example, mm^{3}, \ cm^{3} or m^{3}.

For example,

The volume of this cube is

How to calculate the volume of a cube

In order to calculate the volume of a cube:

Substitute the values into the formula.

Work out the calculation.

Write the answer and include the units.

Explain how to calculate the volume of a cube

Volume and surface area of a cube worksheet

Get your free volume and surface area of a cube worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Volume of a cube examples

Example 1: volume of a cube.

Work out the volume of the cube.

Write down the formula.

2 Substitute the values into the formula.

Here the sides of the cube are 6cm.

3 Work out the calculation.

4 Write the answer and include the units.

The measurements are in cm therefore the volume will be in cm^{3}.

Example 2: volume of a cube

Work out the volume of this cube.

Here the edges are 7cm.

\text{Volume }=7^{3}

\text{Volume }=343 \mathrm{~cm}^{3}

Example 3: volume of a cube – different units

Notice here that one of the units is in cm whilst the other is in m. We need all the units to be the same to calculate the volume. This is a cube so we know all the edges are the same length, so we can easily change m to cm, \ 0.6 \ m = 60 \ cm.

\text{Volume }=60^{3}

\text{Volume }=216000 \mathrm{~cm}^{3}

How to work out a missing length given the volume

Sometimes we are given the volume of a cube and need to work out the length of an edge.

We can do this by substituting the volume into the formula for the volume of a cube and then solving the equation that is formed. This will give us the length of the sides of a cube.

• Write down the formula. \text{Volume }=a^{3}

Substitute the values that you do know into the formula.

• Solve the equation.

Finding the length given the volume examples

Example 4: find the length of a cube given the volume.

The volume of the cube is 64 \ cm^{3}. Work out the length of the cube.

Solve the equation .

Since the measurement for the volume in this question was in cm^{3}, the length will be in cm.

a=4 \ \mathrm{~cm}

Example 5: find the length of a cube given the volume

The volume of the cube is 1000 \ m^{3}. Work out the length of each side, x.

Since the measurement for the volume in this question was in m^{3}, the length will be in m.

x=10 \ \mathrm{~m}

Common misconceptions

• Missing/incorrect units

You should always include units in your answer.  Volume is measured in units cubed (e.g. mm^{3}, \ cm^{3}, \ m^3 etc).

• Calculating with different units

You need to make sure all measurements are in the same units before calculating volume. For example, you can’t have some in cm and some in m.

• Dividing by three rather than cube rooting

If you are given the volume of a cube and you need to find the side length, remember the inverse of cube is cube root, not divide by 3.

For example, if the volume of a cube is 27 \ cm^{3}, the side length is \sqrt[3]{27}=3 \ \mathrm{cm} (not 27 \div 3 ).

Related lessons

Surface area of a cuboid is part of our series of lessons to support revision on cuboid. You may find it helpful to start with the main cuboid 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:

• Volume of a cuboid
• Surface area of a cuboid
• Surface area of a cube

Practice volume of a cube questions

1. Work out the volume of the cube.

2. Work out the volume of the cube.

3. Work out the volume of this cube.

Notice here that one of the units is in cm whilst the other is in m . We need all the units to be the same to calculate the volume.

This is a cube, we know all the edges are the same length, so we can easily change m to cm, \ 0.4 \ m = 40 \ cm.

4. The volume of a cube is 343 \ cm^{3}. Find the length of the side.

5. The volume of the cube is 125 \ m^{3}. Work out the length of each side x.

6. The volume of a cube is 64 \ m^{3}. Work out the length of each side. Give your answer in cm.

Volume of a cube GCSE questions

1. Here is a cube.

2. (a) This sculpture is formed by placing one cube on top of another. Work out the total volume of the sculpture.

(b) The density of the material used to make the sculpture is 2.7 \ g/cm^{3}. Work out the total mass of the sculpture. Give your answer in kg, to the nearest kg.

You may use \text{density }= \frac{\text{mass}}{\text{volume}} .

Volume of bottom cube = 60 \times 60 \times 60=216000 \ cm^3

Volume of top cube = 35 \times 35 \times 35=42875 \ cm^3

Total volume = 216000+42875=258875 \ cm^3

698962.5 \ g=698.9625 \ kg=699 \ kg (nearest kg )

3. A cube has a volume of 343 \ m^{3}. Find the length of the cube. Give your answer in cm.

x^{3}=343 or x=\sqrt[3]{343}

Learning checklist

You have now learned how to:

• Understand and apply the formula to calculate the volume of cubes
• Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3D

The next lessons are

• Triangular prism

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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

Volume of a cube.

The volume of a cube is the space it takes up in the three-dimensional plane. It is measured in cubic units such as m 3 , cm 3 , mm 3 , ft 3 , or in 3 . The volume of a cube determines how big it is.

With Edge Length

The basic or common formula to determine the volume of a cube is:

Let us solve some examples involving the above formula.

Find volume of a 5 x 5 cube.

As we know, The edge length determines the size of a cube, Volume ( V ) = a 3 , here a = 5 units = 5 3 = 125 cubic units

What is the volume of a 9 inch cube?

As we know, Volume ( V ) = a 3 , here a = 9 in = 9 3 = 729 in 3

Calculate the volume of a 40ft high cube container.

As we know, The length, breadth, and height of a cube is same, Volume ( V ) = a 3 , here a = 40 ft ∴ V = 40 3 = 64000 ft 3

With Diagonal

The formula to get the volume of a cube when the diagonal is known is:

Here we will use the equation of the diagonal to derive the formula of volume of a cube with diagonal.

Length of Space Diagonal (d) = ${\sqrt{3}a}$ — (1)

The equation for volume of a cube with edge length is V = a 3 , — (2)

Now, replacing the value of a in eqn (1)

a = d/√3, from (1)

V = a 3 = (d/√3) 3 replacing the value of a from (1)

= d 3 /(√3 × √3 × √3)

∴ V = ${\dfrac{\sqrt{3}}{9}d^{3}}$

(V = √3d 3 /9)

Calculate the volume of a cube given its diagonal 3 cm.

As we know, Volume ( V ) = ${\dfrac{\sqrt{3}}{9}d^{3}}$, here d = 3 cm ∴ V = √3 × 3 3 /9 = 5.2 cm 3

finding the EDGE of a cube when the VOLUME is known

Calculate the side length of a cube given its volume 343 cubic millimeters.

Side length of a cube is actually its edge length. Oftentimes the edge is termed as side length. So, we will use an alternative formula to find the edge with the volume, ${a=\sqrt[3] {V}}$, here V = 343 mm 3 ${a=\sqrt[3] {343}}$ = 7 mm

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Volume of a Cube

Learn how to find the volume of a cube., volume of a cube lesson, cube volume formula.

The formula for volume of a cube is given as:

Where V is the volume and a is the edge length.

Volume of a Cube Example Problems

Let's go through a couple of example problems together to practice finding the volume of a cube.

Example Problem 1

Find the volume of a cube with an edge length of 5.

• We can simply plug the edge length into the formula. This gives us:
• V = (5) 3 = 125
• The volume of the cube is 125.

Example Problem 2

The volume of a cube is 343. What is the edge length?

• Let's plug the volume into the formula and then solve for the edge length a .
• The edge length is 7.

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