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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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Problem Solving with Cuboids

October 14, 2022.

In this lesson, five problems link the volume of cuboids to:

• 3D coordinates
• Standard form
• Setting up and solving equations
• Converting between metric units

Throughout the lesson, I asked students to sketch the diagrams so they could label the critical information. Then, I encourage them to work together to share their approaches to each problem and question each other. While most calculations are reasonably simple, I remind students to use a calculator to focus on problem-solving rather than arithmetic.

I recommend teaching this lesson as part of the schemes of work on  perimeter, area and volume  in key stage 3 or  volume and surface area  in Key Stage 4.

Problem Solving with Cuboids and 3D Coordinates

Finding a length when given the volume and area is a common question when learning about cuboids. This question takes the idea up another level by challenging students to find the cross-section area from two 3D coordinates and use the calculated length to find the third coordinate.

If students struggle to get started, I encourage them to sketch the cuboid without the grid and note the cross-sectional area and length FE.

Next, I prompt them to label coordinates C and F on the diagram to find the area of face BFGC.

Problem Solving with Cuboids and Ratio

I’ve yet to find a topic that can not be linked to ratio in sIn this question, students consider which two parts of a three-part ratio combine to give the smallest area.

Most students correctly identify the numbers 2 : 3 but often struggle knowing how to use them to give the area of 150 cm2. If this happens, I encourage them to think about equivalent ratios, for instance, 4 : 6 or 6 : 9 and so on.

Problem Solving with Cuboids and Unknowns

In my experience, because there is limited information, students find this question the most challenging. Therefore, I encourage them to begin by finding the length of the pink cube.

Next, they need to consider what changes and remains the same without the blue cuboid. When students realise both shapes have the same height, they can find the blue cross-sectional area. From this, go on to work out x.

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About Mr Mathematics

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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Volume of a cube

Here you will learn about the volume of a cube, including how to calculate the volume of a cube within mathematical problems and within real-world contexts.

Students will first learn about volume of a cube as part of measurement and data in 5th grade and extend their learning as part of geometry in 6th grade.

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, you can use the volume of a cube formula, \text {Volume }=a^{3}.

Volume is measured in cubic units. For example, cubic inches (in^3), cubic meters (m^3), or cubic centimeters (cm^3).

For example,

The volume of this cube is,

volume = a^3

volume = 8^3

volume = 512 \, cm^3

The length, width, and height of the cube are multiplied together to find the total volume.

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

• Grade 5 – Measurement and Data (5.MD.3) Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
• Grade 5 – Measurement and Data (5.MD.4) Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
• Grade 5 – Measurement and Data (5.MD.5) Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, for example, to represent the associative property of multiplication. b. Apply the formulas V = l \times w \times h and V = b \times h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
• Grade 6 – Geometry (6.G.2) Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l \times w \times h and V = b \times h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

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.

[FREE] Volume Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of volume. 10+ questions with answers covering a range of 6th, 7th and 8th grade volume topics to identify areas of strength and support!

Volume of a cube examples

Example 1: volume of a cube.

Find the volume of the cube.

Write down the formula.

\text{Volume }=a^{3}

2 Substitute the values into the formula.

Here, the sides of the cube are 6 \, cm.

\text{Volume }=6^{3}

3 Work out the calculation.

\begin{aligned} \text{Volume} &=6 \times 6 \times 6\\\\ &=216 \end{aligned}

4 Write the answer and include the units.

The measurements are in centimeters. Therefore, the volume will be in cubic centimeters.

\text{Volume }=216 \mathrm{~cm}^{3}

Example 2: volume of a cube

Find the volume of this cube.

Here, the edges are each 7 \, in.

\text{Volume }=7^{3}

\begin{aligned} \text{Volume} &=7 \times 7 \times 7\\\\ &=343 \end{aligned}

The measurements are in inches. Therefore, the volume will be in cubic inches.

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

Example 3: volume of a cube – different units

Notice here that one of the units is in centimeters while the other is in meters. You need all the units to be the same to calculate the volume.

This is a cube, so you know all the edges are the same length, so you can easily change meters to centimeters.

0.6 \mathrm{~m}=60 \mathrm{~cm}

\text{Volume }=60^{3}

\begin{aligned} Volume &=60×60×60 \\\\ &=216,000 \end{aligned}

\text{Volume }=216,000 \mathrm{~cm}^{3}

Example 4: volume of a cube – different units

Notice here that one of the units is in meters, one is centimeters, and another is in millimeters. You need all the units to be the same to calculate the volume.

This is a cube, so you know all the edges are the same length, so you can easily change the centimeters and the millimeters to meters.

\begin{aligned} & 15,000 \mathrm{~mm}=15 \mathrm{~m} \\\\ & 1,500 \mathrm{~cm}=15 \mathrm{~m} \end{aligned}

\begin{aligned} \text { Volume } & =15 \times 15 \times 15 \\\\ & =3,375 \end{aligned}

The measurements are in meters. Therefore, the volume will be in cubic meters.

\text{ Volume }=3,375 \mathrm{~m}^3

Example 5: volume of a cube – word problem

Anna has a Rubik’s cube. Each edge of the Rubik’s cube is 5.8 \, cm. What is the volume of the Rubik’s cube?

Here, the edges are each 5.8 \, cm.

\text{ Volume }=5.8^3

\begin{aligned} \text { Volume } & =5.8 \times 5.8 \times 5.8 \\\\ & =195.112 \end{aligned}

\text { Volume }=195.112 \mathrm{~cm}^3

Example 6: volume of a cube – word problem

Grant is moving to a new house and needs to buy boxes to pack his belongings into. He wants to know the volume of the cube-shaped box shown below. The length of the side of the box is 20 inches. Help him determine the box’s volume.

Here, the edges are each 20 \, in.

\text{Volume }=20^{3}

\begin{aligned} \text { Volume } & =20 \times 20 \times 20 \\\\ & =8,000 \end{aligned}

\text{Volume }=8,000 \mathrm{~in}^3

Teaching tips for volume of a cube

• Be sure to give students a wide variety of practice problems within their worksheets that incorporate cubes and other three-dimensional objects found in the real world. This will give students a deeper understanding of the concept of volume.
• Students should fully understand why the volume formula works with a cube – they need to understand why multiplying 3 of the cubes edges (length, width and height) gives us the cube’s volume. Once they have a strong understanding of the formula, they can easily use it to find the volume of any cube.

Easy mistakes to make

• Forgetting to include the units in your answer or writing the incorrect units. You should always include units in your answer. Volume is measured in cubic units. (For example, mm^3, cm^3, m^3, etc).
• Not converting all measurements to the same unit You need to make sure all measurements are in the same units before calculating volume. For example, you can’t have some measurements in centimeters and some in meters.
• Confusing volume of a cube with surface area of a cube Volume is the space inside a cube and it is a three-dimensional measurement. Surface area is the total area of each of the cube’s square faces and it is a two-dimensional measurement.

Related volume lessons

• Volume of a cylinder
• Volume of a hemisphere
• Volume of a sphere
• Volume formula
• Volume of a prism
• Volume of a cone
• Volume of a triangular prism
• Volume of a rectangular prism
• Volume of square pyramid
• Volume of a pyramid

Practice volume of a cube questions

1. Find the volume of the cube.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text{Volume } &=3^{3} \\\\ &=3 \times 3 \times 3 \\\\ &=27 \mathrm{~cm}^{3} \end{aligned}

2. Find the volume of the cube.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text{Volume }&=0.5^{3}\\\\ &=0.5 \times 0.5 \times 0.5\\\\ &=0.125 \mathrm{~m}^{3} \end{aligned}

3. Find the volume of this cube.

This is a cube, which means all the edges are the same length, so you can easily change meters to centimeters, 0.4m = 40 \, cm.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text{Volume }&=40^{3}\\\\ &=40 \times 40 \times 40\\\\ &=64,000 \end{aligned}

\text{Volume }=64,000 \mathrm{~cm}^{3}

4. Kara has 5 sugar cubes to put into her coffee. Each sugar cube has side lengths of 13 \, mm. What is the total volume of all 5 sugar cubes?

First, you need to find the volume of one sugar cube.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text { Volume } & =13^3 \\\\ &= 13 \times 13 \times 13 \\\\ &= 2,197 \mathrm{~mm}^3 \end{aligned}

Since each sugar cube has the same side lengths, you can multiply the volume of one sugar cube by 5 to find the total volume of all 5 sugar cubes.

2,197 \mathrm{~mm}^3 \times 5=10,985 \mathrm{~mm}^3

5. Vera has two boxes, box A and box B, which are shown below. How much greater is the volume of box B than the volume of box A?

First, you need to find the volume of box A and the volume of box B. Then you need to find the difference between the two.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text { Volume } & =12^3 \\\\ & = 12 \times 12 \times 12 \\\\ & = 1,728 \mathrm{~in}^3 \end{aligned}

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text { Volume } & =16^3 \\\\ &= 16 \times 16 \times 16 \\\\ &= 4,096 \mathrm{~in}^3 \end{aligned}

4,096-1,728=2,368

The volume of box B is 2,368 \mathrm{~in}^3 greater than the volume of box A.

6. This sculpture is formed by placing one cube on top of another. Find the total volume of the sculpture.

To find the total volume of the sculpture, you need to find the volume of the top cube and the volume of the bottom cube, then add the volumes together.

Volume of bottom cube: 60 \times 60 \times 60=216,000 \mathrm{~cm}^3

Volume of top cube: 35 \times 35 \times 35=42,875 \mathrm{~cm}^3

Total volume: 216,000+42,875=258,875 \mathrm{~cm}^3

Volume of a cube FAQs

A cube is a three-dimensional shape with 6 square faces. The edges of the cube are of equal length.

To find the volume of a cube, with side length a, you can use the volume of a cube formula, \text { Volume }=a^3. This is the same as multiplying length times width times height.

The volume of a cube formula is \text { Volume }=a^3.

The next lessons are

• Surface area
• Pythagorean theorem
• Trigonometry

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Volume of Cuboid

The volume of cuboid is the quantity that is used to measure the space in a cuboid. A cuboid is a three-dimensional shape that can be seen around us very often. The term volume is used in measuring the capacity of any shape based on its dimensions such as length, breadth, and, height. To calculate the volume of a cuboid, a formula specific to the shape of a cuboid will be used. In this section, we will be learning the formula to calculate the volume of cuboid and solve a few examples to understand the concept better.

What is the Volume of Cuboid?

The volume of the cuboid is the measure of the space occupied within a cuboid. The cuboid is a three-dimensional shape that has length, breadth, and height. If we have a rectangular sheet and we go on stacking such sheets, we will end up getting a shape that has some length, breadth, and height. This stack of sheets looks like a shape that has 6 faces, 12 edges, and 8 vertices hence giving us the shape of a cuboid. The unit of volume of cuboid is given as the (unit) 3 . The metric units of volume are cubic meters or cubic centimeters while the United States Customary System (USCS) units of volume are, cubic inches or cubic feet. The volume of the cuboid depends on the length, breadth, and height of the cuboid, hence changing any one of those quantities changes the volume of the shape.

Volume of Cuboid Formula

The formula for the volume of the cuboid can be derived from the concept explained on rectangular sheets. Let the area of a rectangular sheet of paper be 'A', the height up to which they are stacked be 'h' and the volume of the cuboid be 'V'. Then, the volume of the cuboid is given by multiplying the base area and height. The volume of cuboid = Base area × Height The base area for cuboid = l × b Hence, the volume of a cuboid, V = l × b × h = lbh

How to Calculate Volume of Cuboid?

The volume of a cuboid is the space occupied inside a cuboid. If all the three dimensions of a cuboid get equal, it becomes a cube . The volume of the cuboid can be calculated using the formula of the volume of the cuboid. The steps to calculate the volume of a cuboid are:

• Step 1: Check if the given dimensions of cuboids are in the same units or not. If not, convert the dimensions into the same units.
• Step 2: Once the dimensions are in the same units, multiply the length, breadth, and height of the cuboid.
• Step 3: Write the unit in the end, once the value is obtained.

Let us take an example to learn how to calculate the volume of a cuboid using its formula.

Example: Find the volume of the cuboid having a length of 7 inches, breadth of 5 inches, and height of 2 inches. Solution: As we know, the volume of a cuboid, V = lbh Here, length l = 7 inches, breadth b = 5 inches and height h = 2 inches Thus, volume of cuboid, V = lbh = (7 × 5 × 2) in 3 ⇒ V = 70 in 3 \(\therefore\) The volume of cuboid is 70 in 3 .

Volume of Cuboid Examples

Example 1: If the dimensions of a cuboidal fish aquarium are, 30 inches, 20 inches, and 15 inches. Can you determine the volume of the fish aquarium? Solution: As we know, the fish aquarium is of cuboidal shape. Hence, the dimensions of the fish aquarium are: Length of the aquarium = 30 in Width of the aquarium = 20 in Height of the aquarium = 15 in

The volume of the aquarium is given as: Volume = Length × Width × Height ⇒ Volume = 30 × 20 × 15 in 3 = 9000 in 3 ∴ The volume of the cuboidal fish aquarium is 9000 cubic inches.

Example 2: What will be the length of the cuboid if its volume is 3000 in 3 , breadth is 15 inches and height is 10 inches? Solution: As we know, volume of a cuboid is given as Volume = Length × Breadth × Height. The given dimensions for cuboid are: Volume = 3000 in 3 Breadth = 15 in Height = 10 in Let the length of cuboid is x inches.

Hence, the volume of the cuboid will be: Volume = Length × Breadth × Height ⇒ Volume = x × 15 × 10 = 3000 in 3 ⇒ x = (3000/(15 × 10)) = 20 in ∴ The length of cuboid is 20 inches.

Example 3: Find the volume of cuboid if the length is 10 in, breadth is 20 in, and height is 30 in.

Solution: volume of a cuboid is given as Volume = Length × Breadth × Height. The given dimensions for cuboid are: Length = 10 in Breadth = 20 in Height = 30 in ⇒ Volume = 10 × 20 × 30 in 3 = 6000 in 3 ∴ The volume of the cuboid is 6000 cubic inches.

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Practice Questions on Volume of Cuboid

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FAQs on Volume of Cuboid

What do you mean by volume of cuboid.

The volume of a cuboid is the space that is enclosed within a cuboid. For example, in order to fill water in an aquarium, we must know its volume.

How to Find the Volume of Cuboid?

The volume of a cuboid is calculated by multiplying its length, width, and height. For example, the volume of a cuboid of length, width, and height 2 inches, 3 inches, and 4 inches is given as Volume = length × width × height = 2 × 3 × 4 = 24 inch 3

What Is the Formula for the Volume of Cuboid?

The formula of volume of a cuboid is = Length × Width × Height. The formula for the volume of cuboid is deduced by stacking rectangular sheets one over another thereby giving us three parameters in the formula length, width, and height.

If the Units of Dimensions of a Cuboid Are Different, Then How to Find the Volume of Cuboid?

If the units of the given dimensions of a cuboid are different, then first we would need to change the units of dimensions of any two dimensions in the unit of the third dimension. After that, multiply all three dimensions known to us, to calculate the volume of the cuboid.

Does the Order of Height, Width, and Length Matter While Calculating the Volume of Cuboid?

No, the order of height, width, and length does not matter while finding a cuboid's volume because we need to multiply all three quantities for determining it. As multiplication is associative, hence, no matter in whichever order dimensions are multiplied the volume of the cuboid remains the same.

How Does the Volume of Cuboid Change When the Length of Side Is Halved?

The volume of the cuboid gets half when the length of the side is halved as l = l/2. As, volume of cuboid = length × width × height = (l/2)× b × h = (lbh)/2 = volume/2. Thus, the volume of the cuboid gets halved as soon as its length gets halved.

Volume of a Cuboid

A cuboid is a 3 dimensional shape. To work out the volume we need to know 3 measurements.

The volume is found using the formula:

Volume = Length × Width ×  Height

Which is usually shortened to:

V = l × w × h

Or more simply:

In Any Order

It doesn't really matter which one is length, width or height, so long as you multiply all three together.

Example: Lengths in meters (m):

The volume is:

10 m × 4 m × 5 m = 200 m 3

It also works out the same like this:

4 m × 5 m × 10 m = 200 m 3

Try It Yourself

Cuboid Calculator

What is a cuboid, how to use our cuboid calculator, how to calculate the volume of a cuboid, how to calculate the surface area of a cuboid, similar calculators.

Are you struggling with cuboid calculations? Our cuboid calculator is just the tool you need to help solve those math problems. Keep reading to learn:

• What is a cuboid;
• How to use our cuboid calculator;
• The surface area and volume of a cuboid formula; and
• How to manually calculate the surface area and volume of a cuboid.

A cuboid is a three-dimensional shape that has six rectangular faces. A cuboid's length, width, and height are of different measurements. The corners of these faces form right angles. Cuboids have eight vertices and twelve edges.

Our cuboid calculator calculates the surface area of a cuboid as well as the volume. To calculate either of these two unknowns, you simply need to enter the length ( l ), width ( w ), and height ( h ) of the shape, and the result will be generated in real-time.

As a bonus, you will also be able to see the diagonal of the shape.

To calculate the volume of a cuboid , we use the following formula:

So if we have the following problem: Find the volume of a cuboid whose length is 12 cm, width is 9 cm, and height is 10 cm.

Using the above formula, we can then say:

We calculate the surface area of a cuboid by finding the sum of the area of its six faces. To calculate the surface area of a cuboid ( SAC ), we use the following formula:

So if we need to find the total surface area of a cuboid whose length is 8 cm , width is 7 cm , and height is 6 cm , using the above, we begin by substituting the values:

Are you interested in other similar calculators? Be sure to check out the ones below:

• Rectangular prism calculator ;
• Cuboid surface area calculator ;
• Cuboid volume calculator .

What is the difference between a cube and a cuboid?

The main difference between a cube and a cuboid is that the length, width, and height of a cube are of equal measurements. On the other hand, a cuboid has length, width, and height that are of different sizes.

How do I calculate the volume of a cuboid?

To calculate the volume of a cuboid you need to:

• Get the length, width, and height.
• Put the dimensions into the formula: volume = (l × w × h) cubic unit.
• Do the calculation to find the answer.

Dividing exponents

Meat footprint, surface area of a cube.

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Volume of Cuboid – Definition, Formula, Derivation, Examples, FAQs

What is the volume of a cuboid, volume of a cuboid formula, how to calculate the volume of a cuboid, solved examples of volume of a cuboid, practice problems for volume of a cuboid, frequently asked questions about volume of cuboid.

The volume of a cuboid is the amount of space occupied by the cuboid. It is calculated by multiplying the length, width, and height of the cuboid.

In geometry , a cuboid is a geometric solid with 6 faces, 12 edges, and 8 vertices.. The opposite faces of every cuboid are equal.

What do you mean by the volume of a cuboid? Just like area denotes the space occupied by an object on a 2D plane, the volume represents the space occupied by a solid in the 3D space. A cuboid is a three-dimensional solid. The total 3D space occupied by a cuboid is its volume.

A common real-life example of the volume of a cuboid is the amount of water that completely fills a cuboid-shaped aquarium.

The volume of cuboid is a product of its length, breadth, and height. 

The volume of a cuboid formula is written as

Volume of a cuboid $=$ length $×$ breadth $×$ height

Volume of a cuboid $= l \times b \times h$    

Volume is measured in cubic units.

Step 1: Note down the dimensions of the given cuboid as $l = length,\; b = breadth,$ and $h = height$. 

Step 2: Check whether they are all in the same unit or not. If we come across length, breadth, or height in different units, convert them into the same unit. 

Step 3: Substitute the values l, b, and h in the volume formula $V = l \times b \times h$. 

The resultant value will be the volume of a cuboid. It will be written with cubic units.

Volume of a Cuboid Prism

Cuboid prism or rectangular prism are just other names for a cuboid. A cuboid prism has a rectangular cross-section. It is called a right prism when the angles between its sides (lateral faces) and the base are right angles . Its top surface and the corresponding bottom will be identical. 

Therefore, using the same volume of cuboid formula, we can calculate the volume of a cuboid prism:

Volume of a cuboid prism $= l \times b \times h$   (cubic units)

If $l = b = h$, a cuboid becomes a cube. Its volume is given by $(side)^{3}$.

Volume of cube and cuboid are both expressed in $unit^{3}$.

Derivation of the Volume of a Cuboid

Interestingly, we can also calculate the volume of cuboid if we know its base area and height. Suppose the area of the cuboid’s rectangular face is A and the height of the cuboid is “h.” 

Since volume is the space occupied, the mathematical expression will be as follows 

Volume of a rectangular prism $=$ Base area $\times$ Height

$V = A \times h$ ————— (i)

As we know that the area of a rectangular surface can be calculated using the following formula:

$Area = length \times breadth$   

$A = l \times b$ —————- (ii)

Substituting equation (ii) in equation (i), we get the following:

$V = A \times h$

$V = (l \times b) \times h$

Thus, we get the formula of a cuboid as follows:

$V = l \times b \times h$

Volume of Cuboid Using Unit Cubes

A unit cube is a cube whose each side is 1 unit. The volume of a cuboid can also be defined as the number of unit cubes that fit perfectly into the cuboid.

You can see that 8 unit cubes perfectly fit into the given cuboid.

Thus, volume of cuboid $= 8$ cubic units

Facts about Volume of Cuboid

• Cuboid is also known as a rectangular prism, rectangular box, rectangular parallelepiped, or a rectangular brick!
• Total Surface Area of Cuboid $= 2(lb + bh + hl)$

In this article, we learned how to find the volume of a cuboid, its formula, derivation, and examples. Now, we will solve a few examples and practice problems for revision.

1. What is the volume of the given cuboid?

Solution: 

$l = 6$ inches

$b = 2$ inches

$h = 4$ inches

Volume of cuboid $= l \times b \times h$

Volume of cuboid $= 6 \times 4 \times 2$

Volume of cuboid $= 48\; inches^{3}$

2. The dimensions of the cuboid-shaped aquarium are: $h = 5$ inches, $l = 10$ inches, and $b = 8$ inches. What is the volume of the cuboid?

$l = 10$ inches

$b = 8$ inches

$h = 5$ inches

Volume of cuboid $= 10 \times 8 \times 5$

Volume of cuboid $= 400\; inches^{3}$

3. Find the length of the cuboid, if its volume is $24\; inches^{3}$ . Given: breadth $= 6$ inches and height $= 2$ inches.

Let the length of the cuboid be l.

$24 = l \times 6 \times 2$

$24 = l \times 12$

Therefore, the length of the cuboid is 2 inches.

Volume of Cuboid - Definition, Formula, Derivation, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the volume of the given cuboid?

A cuboid-shaped swimming pool is 20 feet long, 10 feet deep, and 8 feet wide. What is the volume of the swimming pool?

The formula to find the volume of a cuboid with length l, breadth b, and height h is ___, what is the volume of the given rectangular prism.

What is the volume of a cube formula?

Since all sides of a cube are the same, the volume of a cube is equal to the cube of its side. Mathematically, $V = side^{3}$.

How to convert the volume of a cuboid in $inch^{3}$ to $feet^{3}$ ?

To convert a given volume in $inch^{3}$, we will divide the volume value by 1728 to get its $feet^{3}$ equivalent.

How will the volume of a cuboid change when we double the length of its side?

As the volume of cuboid $= length \times width \times height$, we will double the length. 

So, $2l \times b \times h = 2$ volume. 

Thus, the volume of the cuboid will be doubled when we double the length of its side.

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Worksheet on Volume of a Cube and Cuboid

We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.

1. Fill in the blanks:

(i) The volume of a rectangle box (or cuboid) = ________ × ________ × ________

(ii) The volume of a cube = ________ × ________ × ________

(iii) The volume of a 1 cm cube is ________

2. Find the volume of the cube whose each edge is:

3. Find the volume of the cuboid whose dimensions are:

(i) length = 5 m, breadth = 4 m, height = 3 m

(ii) length = 48 cm, breadth = 36 cm, height = 24 cm

(iii) length = 12 m, breadth = 5 m, height = 4 m

Worksheet on  Word Problems on Volume of a Cube and Cuboid:

4. A cube with an edge of 7 cm and a cuboid measuring 7 cm × 4 cm × 8 am are kept on a table. Which shape has more volume?

5. A cuboid is 9 cm long, 5 cm broad and 4 cm high and a cube has an edge of 5 cm. Which one has greater volume?

6. What is the volume of a brick of ice-cream with length 25 cm, breadth 10 cm and height 8 cm?

7. A brick measures 15 cm in length, 8 cm in breadth and 5 cm in height. How many bricks will be used to make a wall of length 15 m, breadth 10 m and height 8 metres?

8. A pond is 50 m long, 30 m wide and 2 m deep. Find the capacity of the pond in cubic metre.

9. Harry’s book shelf is 40 cm long, 50 cm wide and 90 cm high. What is the volume of the book shelf?

10. Find the volume of the following rectangular solids:

11. Complete the given table:            

Answers for the worksheet on volume of a cube and cuboid are given below.

1. (i) length × breadth × height

(ii) Side × Side × Side

(iii) 1 cu cm

2. (i) 125 cu cm

(ii) 64 cu cm

(iii) 216 cu cm

3. (i) 60 cu m

(ii) 41472 cu cm

(iii) 240 cu m

6. 2000 cu. cm

8. 3000 cu m

9. 180000 cu. cm

10. (i) 84 cu cm

(ii) 24 cu cm

(iii) 240 cu cm

(iv) 1200 cu cm

(v) 765 cm 3

(vi) 253.125 cm 3

11. (i) 160 cu cm

(ii) 3 cm, 3 cm

(iii) 11 cm

Units of Volume .

Practice Test on Volume.

Worksheet on Volume of a Cube and Cuboid .

Worksheet on Volume.

5th Grade Geometry Page 5th Grade Math Problems 5th Grade Math Worksheets From Worksheet on Volume of a Cube and Cuboid to HOME PAGE

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5th Grade Volume Worksheets

Welcome to our 5th Grade Volume Worksheets page.

Here you will find our collection of worksheets to introduce and help you learn about volume.

These worksheets will help you to understand and practice how to find the volume of rectangular prisms and other simple shapes.

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Here are the instructions how to enable JavaScript in your web browser .

Volume of Rectangular Prisms

On this webpage you will find our range of worksheets to help you work out the volume of simple 3d shapes such as rectangular prisms.

They are also very useful for introducing the concept of volume being the number of cubes that fill up a space.

These sheets are graded from easiest to hardest, and each sheet comes complete with answers.

Using these sheets will help your child to:

• learn how to find the volume of simple 3d shapes by counting cubes;
• learn how to find the volume of rectangular prisms by multiplying length x width x height
• practice using their knowledge to solve basic volume problems.

What is Volume?

• Volume is the amount of space that is inside a 3 dimensional shape.
• Because it is an amount of space, it has to be measured in cubes.
• If the shape is measured in cm, then the volume would be measured in cubic cm or cm 3
• If the shape is measured in inches, then the volume would be measured in cubic inches or in 3

Volume of a Rectangular Prism

• The volume of a rectangular prism is the number of cubes it is made from.
• To find the number of cubes, we need to multiply the length by the width by the height.
• So Volume = length x width x height or l x w x h.
• We could also multiply the area of the base (which is the length x width) by the height.
• So Volume = l x w x h or b x h (where b is the area of the base)

In the example above, the length is 3, the width is 6 and the height is 2.

So the volume is 3 x 6 x 2 = 36cm 3 or 36 cubic cm.

This tells us that there are 36 cm cubes that make up the shape.

We have split our worksheets up into different sections, to make it easier for you to select the right sheets for your needs.

• Section 1 - Find the Volume by Counting Cubes
• Section 2 - Finding the Volume by multiplication
• Section 3 - Match the Volume (multiplication)
• Section 3 - Volume Problem Solving Riddles

5th Grade Volume Worksheets - Counting Cubes

• Volume - Count the Cubes Sheet 1
• PDF version
• Volume - Count the Cubes Sheet 2

5th Grade Volume Worksheets - Find the Volume by Multiplication

The first sheet is supported, the other two sheets are more independent.

You can choose between the standard or metric versions of sheets 2 and 3 (the measurements are the same)

• Find the Volume Sheet 1 (supported)
• Find the Volume Sheet 2 (standard)
• Find the Volume Sheet 2 (metric)
• Find the Volume Sheet 3 (standard)
• Find the Volume Sheet 3 (metric)

5th Grade Volume Worksheets - Match the Volume

• Match the Volume Sheet 1
• Match the Volume Sheet 2

5th Grade Volume Worksheets - Volume Riddles

• Volume Riddles Sheet 5A
• Volume Riddles Sheet 5B

Volume of Rectangular Prisms Walkthrough Video

This short video walkthrough shows several problems from our Find the Volume Sheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video below!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Volume of a Cube/Cuboid/Box Calculators

Each of the pages below includes a handy calculator to help you find the volume of cubes, cuboids and boxes.

• Volume of a Cube Calculator

• Volume of a Box Calculator

Converting Measures Worksheets

Here is our selection of converting units of measure for 3rd to 5th graders.

These sheets involve converting between customary units of measure and also metric units.

• Converting Customary Units Worksheets
• Metric Conversion Worksheets

5th Grade Geometry Worksheets

Here is our selection of 5th grade Geometry worksheets about angles.

The focus on these sheets is angles on a straight line, angles around a point and angles in a triangle.

• 5th Grade Geometry Missing Angles

Area Worksheets

Here is our selection of free printable area worksheets for 3rd and 4th grade.

The sheets are all graded in order from easiest to hardest.

• work out the areas of a range of rectangles;
• find the area of rectilinear shapes.
• Area Worksheets - Rectangles and Rectilinear Shapes
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• Perimeter Worksheets

Here is our selection of free printable perimeter worksheets for 3rd and 4th grade.

• work out the perimeter of a range of rectangles;
• find the perimeter of rectilinear shapes.

All the math practice worksheets in this section support Elementary Math Benchmarks.

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Volume of a Cube

In these lessons, we will learn

• what is volume?
• how to find the volume of a cube.
• how to solve word problems about cubes.
• about nets of a cube.

Related Pages Volume Of Rectangular Prism Volume Formula More Geometry Lessons

Solid geometry is concerned with three-dimensional shapes and their properties.

What is Volume?

The volume of a three-dimensional shape is a measurement of the space occupied by the shape.

Volume is measured in cubic units.

The volume of a unit cube = 1 unit × 1 unit × 1 unit = 1 unit 3 ( Read as one cubic unit )

The volume of a cube with sides 1 cm × 1 cm × 1 cm

Volume = 1 cm × 1 cm × 1 cm = 1 cm 3 ( Read as one cubic cm )

Some important units of conversion for volume are: 1 cm 3 = 1,000 mm 3 1 m 3 = 1,000, 000 cm 3

A cube is a three-dimensional figure with equal edges and six matching square sides.

The figure above shows a cube. The dotted lines indicate edges hidden from your view.

If s is the length of one of its sides, then the volume of the cube is s × s × s

Volume of the cube = s 3

Since the cube has six square-shape sides, the

Surface area of a cube = 6 s 2

Worksheet to calculate volume and surface area of cubes.

How to find the volume of a cube?

Step 1: Find the length of a side Step 2: Substitute into the equation Step 3: Evaluate Step 4: Write the units

Find the volume of a cube with side length of 20 feet.

Word Problems about Cubes

This video gives a word problem about the volume of cubes.

Example: Alex formed the solid as shown below. It is made up of 2-in cubes. What is the volume of the solid?

Example: A shipping company fills cases with boxes that are cubes. They pack 20 boxes of cubes in a case. What are the possible dimensions of the case?

Nets of a Cube

Imagine making cuts along some edges of a cube and opening it up to form a plane figure. The plane figure is called the net of the cube. There are 11 possible nets for a cube as shown in the following figures.

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How to Solve Word Problems of Volume of Cubes and Rectangular Prisms

Today, we're exploring the world of volume, specifically focusing on cubes and rectangular prisms. We're taking it a step further by diving into word problems, a vital skill for applying math to real-world situations!

1. Understanding Volume

Volume is the measure of space occupied by a three-dimensional object. For a cube or rectangular prism, it’s calculated by multiplying the length, width, and height of the object.

2. Solving Volume Word Problems

Word problems involve taking a real-world situation and translating it into mathematical terms. For volume problems, we’re usually given or need to find the dimensions of a shape and calculate the volume.

Step-By-Step Guide to Solving Volume Word Problems

Let’s break down the process:

Step 1: Understand the Problem

First, read the problem carefully to identify the known values (usually dimensions of the shape) and what you need to find (typically the volume).

Step 2: Visualize or Draw the Shape

It’s often helpful to draw the shape and label its dimensions.

Step 3: Use the Volume Formula

For a cube or rectangular prism, the volume is calculated as length x width x height.

Step 4: Solve and Check

Plug in the known values into the volume formula and solve. Then, check your answer to make sure it makes sense in the context of the problem.

For example, let’s consider a problem: “A rectangular box has a length of \(5\ cm\), a width of \(3\ cm\), and a height of \(2\ cm\). What is the volume of the box?”

• Understand the problem: We know the dimensions of the box and need to find the volume.
• Visualize the shape: Draw a rectangular box and label the dimensions.
• Use the volume formula: \(Volume = length\times width\times height = 5\ cm\times 3\ cm\times 2\ cm\).
• Solve and check: The volume is \(30\) cubic cm. This makes sense given the dimensions of the box.

As always, practice is key. Try solving a variety of volume word problems to solidify your understanding.

Keep practicing, keep exploring, and enjoy your mathematical journey!

In this blog post, we’ve explained how to solve word problems involving the volume of cubes and rectangular prisms. We also provided a step-by-step guide to practice this essential skill. Keep practicing, and you’ll master this useful real-world mathematical skill in no time. Happy solving!

by: Effortless Math Team about 9 months ago (category: Articles )

Effortless Math Team

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• (Choice A)   8  cm ‍   long, 1  cm ‍   wide, 3  cm ‍   high A 8  cm ‍   long, 1  cm ‍   wide, 3  cm ‍   high
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• (Choice C)   2  cm ‍   long, 2  cm ‍   wide, 6  cm ‍   high C 2  cm ‍   long, 2  cm ‍   wide, 6  cm ‍   high

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Worked-out Problems on Volume of a Cuboid | How to Find Cuboid Volume?

Students who want to learn the volume of cuboids and cubes can use the Worked-out Problems on Volume of a Cuboid here. Get to see various examples on the cuboid volume in the coming sections. Try to solve the questions and improve your preparation standards. Check out the Cube and Cuboid Word Problems with solutions in the below sections.

Question 1.

Find the volume of a cuboid of length 18 cm, breadth 25 cm, and height 5 cm?

Given that,

Length of cuboid = 18 cm

Breadth of cuboid = 25 cm

Height of cuboid = 5 cm

Cuboid volume = length x breadth x height

Volume = 18 x 25 x 5

Therefore, the volume of a cuboid is 2250 cm³.

Question 2.

If the area of the base and height of the cuboid is 212 cm², 8 cm, calculate cuboid volume?

The base of the base = 212 cm²

Height of a cuboid = 8 cm

Cuboid volume = (Area of the base) x height

Volume = 212 x 8 = 1696 cm³.

Question 3.

Find the volume of the cube whose each side is 16 cm?

Side length of cube a = 16 cm

The volume of the cube V = a³

V = 16³ = 16 x 16 x 16

V = 4096 cm³

Therefore, the cube volume is 4096 cm³.

Question 4.

If the cuboid volume is 512 cm³, its length and height is 8 cm, 7 cm. Find the cuboid breadth?

Cuboid Volume = 512 cm³

Cuboid length = 8 cm

Cuboid height = 7 cm

Cuboid breadth = Volume / (length) x (height)

= 512 / (8 x 7)

= 512 / 56 = 9.142 cm

Therefore, the breadth of cuboid is 9.142 cm.

Question 5.

The length, breadth, and depth of a lake are 15 m, 20 m, 9 m respectively. Find the capacity of the lake in liters?

Length of lake = 15 m

Breadth of lake = 20 m

Depth of lake = 9 m

Capacity of lake = (length) x (breadth) x (depth)

= 15 x 20 x 9 = 2700 m³

1000 liter = 1 m³

Capacity of lake in Litres = 2700 x 1000

= 2700000 litres

Therefore, the capacity of lake is 2700000 litres.

Question 6.

The dimensions of the brick are 25 cm x 8 cm x 10 cm. How many such bricks are required to build a wall of 16 m in length, 20 cm breadth, and 8 m in height?

Length of brick = 25 cm

Breadth of brick = 8 cm

Height of brick = 10 cm

Length of wall = 16 m

Breadth of wall = 20 m

Height of wall = 8 m

Volume of 1 brick = length x breadth x height

= 25 x 8 x 10 = 2000 cm³

Volume of wall = length x breadth x height

= 16 x 20 x 8

= 2560 = 2560 x 100²

Number of bricks required = (2560 x 100²) / 2000

= 1280 x 10 = 12800

So, the required number of bricks are 12800.

Question 7.

External dimensions of a wooden cuboid are 20 cm × 15 cm × 12 cm. If the thickness of the wood is 2 cm all around, find the volume of the wood contained in the cuboid formed.

External length of cuboid = 20 cm

External breadth of cuboid = 15 cm

External height of the cuboid = 12 cm

External volume of the cuboid = (length x breadth x height)

= (20 x 15 x 12) = 3600 cm³

Internal length of cuboid = 20 – 4 = 16 cm

Internal breadth of cuboid = 15 – 4 = 11 cm

Internal height of the cuboid = 12 – 4 = 8 cm

Internal volume of a cuboid = (length x breadth x height)

= (16 x 11 x 8) = 1408 cm³

Therefore, volume of wood = External volume of the cuboid – Internal volume of a cuboid

= 3600 – 1408 = 2192 cm³

∴ Volume of the wood contained in the cuboid is 2192 cm³.

Question 8.

The volume of a container is 1440 m³. The length and breadth of the container are 15 m and 8 m respectively. Find its height?

Length of the container = 15 m

The breadth of the container = 8 m

The volume of the container = 1440 m³

(length x breadth x height) = 1440

15 x 8 x height = 1440

120 x height = 1440

height = 1440/120

height = 12

∴ The height of the container is 12 m.

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SOLVING WORD PROBLEMS INVOLVING VOLUME OF CUBES

Formula : 

Volume of a cube = a 3 cubic units

Problem 1 :

Find the volume of a cube each of whose side is (i) 5 cm (ii) 3.5 m (iii) 21 cm

Volume of cube = a 3

(ii) 3.5 m :

= 42.875 cm 3

(iii) 21 cm :

= 9261 cm 3

Problem 2 :

A cubical milk tank can hold 125000 liters of milk. Find the length of its side in meters.

Volume of cubical milk tank = 125000 liters

1000 liters = 1 m 3 ,

Volume of tank = 125000/1000

a 3  = 125

Problem 3 :

A metallic cube with side 15 cm is melted and formed into a cuboid. If the length and height of the cuboid is 25 cm and 9 cm respectively then find the with of the cuboid.

volume of cuboid = volume of cube

l x w x h = a 3

25 x w x 9 = 15 3

225w = 3375

Divide each side by 225.

Problem 4 :

The sides of two cubes A and B are in the ratio 3 : 5. If the volume of cube A is 729 cm 3 , find the volume of cube B. 

From the ratio 3 : 5, the sides of cubes A and B are 

Volume of cube A = 243 cm 3

(3x) 3 = 729

27x 3  = 729

Divide each side by 27.

x 3  = 27

x 3  = 3 3

Side of cube A = 3(3) = 9 cm

Side of cube B = 5(3) = 15 cm

Volume of cube B : 

= 3375 cm 3

Problem 5 :

If the sides of two cubes are in the ratio 4 : 7, find the ratio of their volumes. 

From the ratio 4 : 7, the sides of two cubes are 

Ratio of their volumes :

= (4x) 3 : (7x) 3

= 64x 3  : 343x 3

Divide both the terms of the ratio by x 3 .

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SOLVING WORD PROBLEMS ON CUBE AND CUBOID

Problem 1 :

Both cuboids below have the same volume. Find the height of cuboid B.

Volume of a cuboid = (length × breadth × height) cubic units

20 × 3 × 15 = 25 × 9 × h

900 = 225 × h

h = 900/225

So, the height of cuboid B is 4 cm.

Problem 2 :

The volume of the cube is twice the volume of the cuboid. Find the length of the cuboid.

Volume of a cube = (side) 3

Volume of a cuboid = y × 4 × 4

Volume of a cuboid = 16y

The volume of the cube is twice the volume of the cuboid.

Volume of a cube = 2 × Volume of a cuboid

216 = 2 × 16y

So, the length of the cuboid is 6.75 cm.

Problem 3 :

The cuboid container below is used to store boxes. Each box is a cube with side length 1m. How many boxes can be stored in the container ?

Volume of a cube = 1

Volume of a cuboid = 5 × 12 × 2

Volume of a cuboid = 120

So, 120 boxes can be stored in the container boxes.

Problem 4 :

The volume of a cuboid is 15000 cm 3 . If the length is 30 cm and the width is 25 cm, find the height of the cuboid.

volume of a cuboid = 15000 cm 3

length of cuboid = 30 cm

width of cuboid =  25 cm

height of the cuboid = ?

15000 = 30 × 25 × h

15000 = 750h

h = 15000/750

So, the height of the cuboid is 20 cm.

Problem 5 :

Shown is a net of a cuboid. Calculate the volume of the cuboid

From the given net diagram,

Length of cuboid = 24 cm

Width = 16 cm

height = 12 cm

Volume of cuboid = length x width x height

= 24 x 16 x 12

= 4608 cm 3

Problem 6 :

Find the surface area of a box with length 12 inches and width and height both 4 inches each.

Surface area = 2(l w + w h + h l)

Length = 12 inches, width = height = 4 inches

= 2 (12 x 4 + 4 x 4 + 4 x 12)

= 2 (48 + 16 + 48)

= 224 inches 2

Problem 7 :

Find the surface area of the shown below.

Area of the top = 32 cm 2

From the given figure, width = 4 cm and height = 6 cm

length x width = 32

length x 4 = 32

length = 32/4 ==> 8 cm

Surface area of rectangular prism = 2(lw + wh + hl)

= 2 (8 x 4 + 4 x 6 + 6 x 8)

= 2(32 + 24 + 48)

Problem 8 :

Find the surface area of cube.

By observing the measures, it is cube.

Side length of cube = 6 inches

Surface area of cube = 6a 2

Problem 9 :

The volume of a cuboid is 15,000 𝑐𝑚 3 . If the length is 30 cm and the width is 25 cm, find the height of the cuboid

Volume of cuboid =  15,000 𝑐𝑚 3

length = 30 cm, width = 25 cm and height = h

Length x width x height = 15,000 𝑐𝑚 3

30 x 25 x h = 15,000

h = (15000) / (30 x 25)

Problem 10 :

The ratio of the width of a cuboid to its height is 4:5. Its width is 40 cm. The ratio of the height to the length is 2:3. Find the volume of the cuboid.

Width of the cuboid = 4x, height = 5x

Width = 40 cm

Width = 4x = 4(10) ==> 40 cm

Height = 5x = 5(10) ==> 50 cm

ratio between height to the length = 2 : 3

2y = height and length = 3y

Applying the value of y, we get

Length = 3y = 3(25) ==> 75 cm

Volume of the cuboid = length x width x height

= 75 x 40 x 50

= 150000 cm 3

Problem 11 :

The two cuboids shown below have the same volume. Calculate the value of 𝑥.

x(x) (10) = 20 x 4 x 8

10x 2 = 640

So, the value of x is 8 cm.

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