problem solving involving volume

Volume Word Problems

Exercise 10, exercise 11, exercise 12, exercise 13, solution of exercise 1, solution of exercise 2, solution of exercise 3, solution of exercise 4, solution of exercise 5, solution of exercise 6, solution of exercise 7, solution of exercise 8, solution of exercise 9, solution of exercise 10, solution of exercise 11, solution of exercise 12, solution of exercise 13.

Calculate the volume (in cubic centimetres) of a prism that is 5 m long, 400 cm wide and 2500 mm high.

A swimming pool is 8 m long, 6 m wide and 1.5 m deep. The water-resistant paint needed for the pool costs 6 dollars per square meter.

1 How much will it cost to paint the interior surfaces of the pool?

2 How many litres of water will be needed to fill it?

A moving company is trying to store boxes in a storage room with a length of 5 m, width of 3 m and height of 2 m. How many boxes can fit in this space if each is 10 cm long, 6 cm wide and 4 cm high?

Calculate the surface area of a regular tetrahedron , octahedron and icosahedron, each with an edge of 5 cm.

Calculate the height of a prism whose base area is 12 dm² and whose capacity is 48 litres.

Calculate the quantity of sheet metal that would be needed to make 10 cylindrical canisters with a diameter of 10 cm and a height of 20 cm.

The height of a cylinder is the same length as the circumference of its base. Its measured height is 125.66 cm. Calculate the surface area and volume of the cylinder:

Four cubes of ice with an edge of 4 cm each are left to melt in a cylinderical glass with a radius of 6 cm. How high will the water rise when they have melted?

The dome of a cathedral has a semispherical form with a radius of 50 m. If the restoration costs 300 dollars per m², what is the total cost of the restoration?

How many square tiles (20 cm x 20 cm) are needed to coat the sides and base of a pool which is 10 m long, 6 meters wide and 3 m deep?

A cylindrical container with a radius of 10 cm and a height of 5 cm is filled with water. If the total mass of the filled container is 2 kg, what is the mass of the empty container?

For a party, Louis has made about 10 conical hats out of cardboard. How much cardboard was used in total if each cap has a radius of 15 cm and a slant height of 25 cm?

A cube with an edge of 20 cm is filled with water. Would this amount of water fit in a sphere with a 20 cm radius?

Calculate the volume (in cubic centimetres) of a prism that is 5 m long, 400 cm wide and 2,500 mm high.

Four cubes of ice with an edge of 4 cm each are left to melt in a cylindrical glass with a radius of 6 cm. How high will the water rise when they have melted?

The dome of a cathedral has a hemispherical form with a radius of 50 m. If the restoration costs 300 dollars per m², what is the total cost of the restoration?

Yes, the amount of water can easily fit inside of the sphere.

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the solution of Exercise No.3 …volume 1 = 30m3

and the last fraction is 30/0.24 = 125

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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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• Earth and space

Solve Word Problems Using Volume

Key concepts.

• Word problems using volume

Solve word problems using volume 

What is volume?  

Space occupied by a 3D shape is known as its volume.  

Let us consider the volume of a shape by multiplying length × width × height. 

A pet shop has a large rectangular bird cage. They need 14 feets for each cage. Find the volume of space required to place all the birds inside the cage. 

Word problems involving volume 

Jerimia built a house out of building blocks. Find the volume of the house Jerimia built. 

Solution:  

Step 1 : Find the volume of each section. 

V = l x w x h  

Step 2 : Find the volume of block on the house. 

V = 5 x 3 x 10 

Step 3 : Find the volume of house. 

V = 20 × 24 × 20 

V = 9,600  

Step 4 : Add to find the total volume: 

V = 150 + 9, 600 

 V = 19,350 

The combined volume is 19, 350 cubic centimeters. 

Example 2 : 

How can Ria find the volume of the china cabinet. Find the height of the top section of the china cabinet? Find the volume of the cabinet? 

Step 1 : Total height of the cabinet is 7 ft 

Bottom part of cabinet height is 3 ft. 

Top part of cabinet height is 7 – 3 = 4 ft 

Step 2 : Find top part of cabinet volume 

V = l   × w   × h  

V = 4 × 1 × 4 

Step 3: Find bottom part of cabinet volume. 

V = l × w   × h  

V = 4 × 2 × 3 

Step 4 : Add two volumes. 

V = 16 + 24 

V = 40 cubic feet 

Volume of China cabinet is 40 cubic feet. 

Example 3 : 

A book is 19 cm wide, 26 cm long and 2 cm thick. There are 8 similar books placed opposite to the top of each other. Find the volume taken up by them. 

Step 1 : Find the volume of Book using below formula. 

Step 2 : Find the volume of one book. 

V = 19 × 26 × 2 

V = 988 cubic centimeters. 

Step 3: We want 8 books volume 

V = 8 × 988 

V= 7, 904 cubic centimeters.  

Volume of 8 books is 7,904 cubic centimeters. 

Example 4:  

Thomas wants to buy a new building. He wants to find the volume of the building. How can he find the total volume of the building? 

Step 2 : Find the volume of back block. 

V = 8 × 4 × 10 

Step 3 : Find the volume of front block. 

V= 6 x 8 x 4 

V= 320 + 192 

V= 704 cubic meters. 

The volume of the building is 704 cubic meters. 

A cargo company is planning to build two adjacent buildings, which are in the shape of a rectangular prism. They want to construct a metal roof inside the building. Find the volume of the space required to construct the metal roof. 

Step 2 : Find the volume of back block of the wing. 

V = 50 x 50 x 10 

V= 25, 000  

Step 3 : Find the volume of front block of the wing. 

V= 75 x 57 x 14 

V = 59, 850 

V= 25, 000 + 59, 850 

V= 84, 850 cubic meters. 

The volume of the wings is 84, 850 cubic meters. 

• Lenin building was designed to have a long first floor with a tower in the middle extending to higher floors, as seen in the image. Find the volume of the building.

• A box has a length of 7 centimeters, a width of 5 centimeters, and a height of 8 centimeters. What is the volume of the box?
• Stanli is using this box to pack books. He needs to know the volume so he can figure out how many books to pack. Find the volume.

• Bob has a rectangular prism gift box what it the volume of bobs gift box?

• Find the volume of below image.

• The figure shows two right rectangular prisms.

• If the volume of the yellow section is 10 cm? and the volume of the blue section is 12cm’, Find the volume of the figure?
• A crate with dimensions 20 inches wide, 20 inches long, and 12 inches high. What is the volume of crate?
• A compost container measures 4 feet long, 3 feet wide, and 2 feet high. Find the volume of the compost container.
• A box is in the shape of a rectangular prism and has a volume of 360 cubic inches. The box has a width of 6 inches and a length of 10 inches. What is the height of the box? 

Concept Map 

What have we learned

• Understanding the volume.
• How to find volume of rectangular prism using the formula (V) =1 x b x h.
• How to find word problems involving volume?
• How to break rectangular prism into two rectangular prisms?
• How to add two rectangular prism volumes.

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SAT (Fall 2023)

Course: sat (fall 2023)   >   unit 10, volume word problems — basic example.

• Volume word problems — Harder example
• Right triangle word problems — Basic example
• Right triangle word problems — Harder example
• Congruence and similarity — Basic example
• Congruence and similarity — Harder example
• Right triangle trigonometry — Basic example
• Right triangle trigonometry — Harder example
• Angles, arc lengths, and trig functions — Basic example
• Angles, arc lengths, and trig functions — Harder example
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Video transcript

Volume Word Problems

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• About us

Volume Word Problems

Test your knowledge and try to solve all of our volume word problems!

Volume of Cubes and Rectangular Prisms

A water tank that is rectangular in shape is 15 feet long, 10 feet wide, and 6 feet deep. What is the volume of the water tank?

900 cubic feet.

Ship container has a length of 40 feet, width 8 feet, and 9.6 feet in height. Find the volume of the container in cubic feet.

40 × 8 × 9.6 = 3,072 cubic feet.

A building cube is a rectangular prism with the same measurement for length, width, and height. If a cube is 8 inches tall, what is its volume?

8 × 8 × 8 = 8³ = 512 cubic inches.

Harry will paint the walls and ceiling of his room that is 16 feet wide by 14 feet long by 8 feet high. What is the surface area of the room?

16 × 14 × 8 = 1,792 cubic feet.

In the store, jewelry is in a glass display case in the shape of a rectangular prism. It is 16 inches long, 11 inches high, and has a volume of 704 cubic inches. Find the width of the rectangular display case.

16 × x × 11 = 704 x = 704 ÷ 176 x = 4 inches width

Jessica wants to send a gift to her grandmother. She needs a box that can fit a volume of 80 cubic inches and has a height of 5 inches. What is the area of the bottom of the box Jessica needs?

80 ÷ 5 = 16²

Alexa was determining the dimensions of her cereal box Friday morning. She read on the box that the total volume was 336 cubic inches. Alexa found it was 3 inches wide and 14 inches high, what was the length?

V = L × W × H 336 = L × 3 × 14 L = 336 ÷ 42 L = 8

solve 10-20 LogicLike's problems and riddles. And how much can you do?

Volume of Cylinders

Chef Mr. Yummy has a very large soup pot shaped like a cylinder that has a height of 14 inches and a radius of 8 inches. How much soup can Mr. Yummy's soup pot hold?

V ≈ 3.14 × 8² × 14 ≈ 2,813.44

Emma says that the volume of a cylindrical vase that has a diameter of 8 cm and a height of 14 cm is about 2,813.44 cm³. Is Emma correct?

V ≈ 3.14 × 4² × 14 ≈ 43.96 No, Emma multiplied the diameter instead of the radius.

The tallest cylindrical aquarium in the world is located in Moscow. Its height is 66.6 feet deep and has a radius of 9.9 feet. Find the volume of the tank.

V ≈ 3.14 × 9.9² × 66.6 V ≈ 20,496.2 cubic feet.

African Bongo Drum has a diameter of 9 inches and is 18 inches deep. Find the volume of this African drum.

9 ÷ 2 = 4.5 V ≈ 3.14 × 4.5² × 18 V ≈ 1,144.5 cubic feet.

If a 5-gallon keg of Pepsi has 25 inches tall and 8.6 inches in diameter, what is the volume of this keg in cubic feet?

8.6 ÷ 2 = 4.3 V ≈ 3.14 × 4.3² × 25 V ≈ 1,451.5 cubic feet.

A cylindrical container with a radius of 10 feet and a height of 5 feet is filled with water. If the total mass of the filled container is 4 kg, what is the mass of the empty container?

V ≈ 3.14 × 10² × 5 V ≈ 1,570

The volume of a cylinder is 441 cubic inches. The height of the cylinder is 9 in. Find the radius of the cylinder to the nearest tenth of an inch.

441 ≈ 3.14 × r² × 9 441 ≈ 28.26 × r² r² ≈ 441 ÷ 28.26 r² ≈ 15.6 inches.

A right circular cylinder has a volume of 628 cubic inches and a height of 8 inches. What is the radius of the cylinder?

628 = 3.14 × r² × 8 628 = 25.12 × r² r² = 628 ÷ 25.12 r² = 25 r = 5

Volume of Composite Figures

The first rectangular prism has a base with a length of 25, a width of 9, and a height of 12. The second prism has a square base with a side of 15 and volume equal to the first prism. The third prism has a height equal to the second prism and a base area equal to the first prism. Find the volume of all three prisms together.

25 × 9 × 12 = 2,700 (V1) 15 × 15 × h = 2,700 (V2) 225 × h = 2,700 h = 2,700 ÷ 225 = 12 V3 = 25 × 9 × 12 = 2,700 2,700 × 3 = 8,100

Using blocks that are 1 cubic foot, Mr.Freeman builds the stage base of 3 blocks wide and 4 blocks long. If he continues to build the stage up, in height, how many more layers will she need to create a stage with a volume of 96 cubic feet?

4 × 3 = 12 96 ÷ 12 = 8 layers.

A cylindrical hole with a diameter of 8 inches is bored through a cub 10 inches on a side. Find the surface area and volume of this solid casting.

V ≈ 3.14 × 4² × 10 ≈ 502.4 V = 10³ = 1,000 1,000 − 502.4 = 497.6 cubic inches.

A cube of side 8 cm is cut into 2 cm cubes. What is the ratio of the surface areas of the original cubes and cut-out cubes?

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