Volume and Surface Area of 3D Figures Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

A rectangular shipping crate has a length of $6$ feet, a width of $4$ feet, and a height of $3$ feet. What is the volume of the crate in cubic feet?

Show answer & explanation

Answer: 72

Answer

72

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Multiplying

6

feet by

4

feet by

3

feet results in exactly

72

cubic feet.

Step-by-Step Solution

1

Recall the volume formula for a rectangular prism.

V = l \times w \times h

To find the volume, we must multiply the length, width, and height of the 3D figure.

2

Substitute the given dimensions into the formula.

V = 6 \times 4 \times 3

The problem states the length is

6

feet, the width is

4

feet, and the height is

3

feet.

3

Multiply the values to find the total volume.

V = 24 \times 3 = 72

Completing the arithmetic gives the final volume in cubic feet.

Key Concept

Volume of a Rectangular Prism

Question 2Question →

A cylindrical canister has a radius of $3$ inches and a height of $5$ inches. Which of the following represents the total volume of the canister in cubic inches?

45\pi

30\pi

75\pi

180\pi

Show answer & explanation

Answer: $45\pi$

Answer

45\pi

The correct answer is

45\pi

. The volume of a cylinder is found using the formula

V = \pi r^2 h

. Squaring the given radius of

3

results in

9

. Multiplying

9

by the given height of

5

yields

45\pi

cubic inches.

Step-by-Step Solution

1

Identify the correct formula for the volume of a cylinder.

V = \pi r^2 h

The question asks for the volume of a cylindrical shape.

2

Substitute the given values into the formula.

V = \pi (3)^2 (5)

The radius (r) is 3 inches and the height (h) is 5 inches.

3

Square the radius according to the order of operations.

3^2 = 9

Exponents must be evaluated before multiplication.

4

Multiply the squared radius by the height and pi.

9 \times 5 \times \pi = 45\pi

This completes the volume calculation.

Key Concept

Volume of a Cylinder

Question 3Question →

Match each 3D solid to its corresponding volume formula.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Rectangular Prism matches with

V = l \times w \times h

; Cube matches with

V = s^3

; Cylinder matches with

V = \pi r^2 h

; Sphere matches with

V = \frac{4}{3} \pi r^3

.

Each 3D solid has a distinct volume formula based on its geometry. Matching them correctly demonstrates fundamental knowledge required before solving applied numerical problems.

Step-by-Step Solution

1

Identify the base shape and defining dimensions for each 3D solid.

Rectangular prisms use length, width, and height. Cubes use a single side length. Cylinders use radius and height. Spheres use only radius.

Volume formulas are derived from the foundational properties and base shapes of the solids.

2

Connect the identified dimensions to the standard algebraic formulas.

All solids are successfully matched to their specific volume equations.

Accurate calculation of volume requires applying the correct formula for the specific geometric figure.

Key Concept

Volume Formulas of Basic 3D Solids

Estimated Time:45s

Question 4Question →

A cube-shaped block has an edge length of $5$ centimeters. What is the total surface area of the block in square centimeters?

25

125

150

900

Show answer & explanation

Answer: 150

Answer

The total surface area of the cube is 150 square centimeters.

The total surface area of a cube is calculated using the formula

SA = 6s^2

, where

s

is the edge length. Substituting

s = 5

gives

6(5)^2

. By following the order of operations, square the

5

first to get

25

, and then multiply by

6

to find the total surface area of

150

square centimeters.

Step-by-Step Solution

1

Identify the formula for the total surface area of a cube.

SA = 6s^2

A cube has 6 identical square faces, and the area of each face is the edge length squared (

s^2

).

2

Substitute the given edge length (

s = 5

) into the formula.

SA = 6(5)^2

This sets up the equation with the specific dimensions of the block.

3

Evaluate the exponent first, following the order of operations.

5^2 = 25

Exponents must be calculated before multiplication.

4

Multiply the area of one face by the number of faces.

6 \times 25 = 150

This provides the combined area of all 6 faces, which is the total surface area.

Key Concept

Volume and Surface Area of 3D Figures

Estimated Time:45s

Question 5Question →

A solid concrete pillar is built in the shape of a rectangular prism. It has a square base with each side measuring $3$ feet, and a total height of $10$ feet. What is the volume of the pillar in cubic feet?

Show answer & explanation

Answer: 90

Answer

90

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Because the base is a square, both the length and the width are

3

feet. Multiplying

3 \times 3 \times 10

yields a total volume of

90

cubic feet.

Step-by-Step Solution

1

Identify the proper formula to calculate the volume of a rectangular prism.

Volume =

l \times w \times h

This is the standard formula used to find the total 3D space occupied by a box-like object.

2

Determine the values of the length, width, and height from the problem description.

Length (

l

) =

3

feet, Width (

w

) =

3

feet, Height (

h

) =

10

feet.

The base is described as a square with a side measuring 3 feet, meaning both the length and width are 3 feet.

3

Multiply the dimensions to calculate the total volume.

3 \times 3 \times 10 = 90

Multiplying the values provides the final volume in cubic units.

Key Concept

The volume of a rectangular prism is found by multiplying its length, width, and height together.

Alternative Method

You can also think of the volume formula as Volume = Area of the Base × height. First, find the area of the square base (

3 \times 3 = 9

square feet). Then, multiply this base area by the height of the pillar (

9 \times 10 = 90

cubic feet).

Estimated Time:45s

Question 6Question →

Match each description of a 3D solid to its corresponding numerical value for volume or total surface area. (Note: All answers are expressed in terms of $\pi$ .)

Click a left item, then click its matching right item

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Answer

The volume of the cylinder matches

48\pi

, the volume of the sphere matches

36\pi

, the total surface area of the cylinder matches

30\pi

, and the surface area of the sphere matches

64\pi

.

Each correct numerical value is found by identifying the specified 3D solid, recalling its corresponding volume or surface area formula, and accurately substituting the given dimensions into the formula.

Step-by-Step Solution

1

Calculate the volume of the cylinder.

V = \pi(2^2)(12) = 48\pi

Apply the cylinder volume formula

V = \pi r^2 h

.

2

Calculate the volume of the sphere.

Radius is

3

.

V = \frac{4}{3}\pi(3^3) = 36\pi

Convert diameter to radius, then apply the sphere volume formula

V = \frac{4}{3}\pi r^3

.

3

Calculate the total surface area of the closed cylinder.

SA = 2\pi(3^2) + 2\pi(3)(2) = 30\pi

Apply the cylinder total surface area formula

SA = 2\pi r^2 + 2\pi rh

.

4

Calculate the surface area of the sphere.

SA = 4\pi(4^2) = 64\pi

Apply the sphere surface area formula

SA = 4\pi r^2

.

Key Concept

Applying appropriate volume and surface area formulas for cylinders and spheres.

Question 7Question →

A cylindrical water tank has a volume of $250\pi$ cubic feet and a height of $10$ feet. The exterior curved side (lateral surface) of the tank needs a new coat of paint, but the flat top and bottom bases will not be painted. What is the total area, in square feet, that needs to be painted?

150\pi

200\pi

100\pi

250\pi

Show answer & explanation

Answer: $100\pi$

Answer

The total area that needs to be painted is

100\pi

square feet.

The area to be painted is the lateral surface area of the cylinder, calculated with the formula

2\pi r h

. To find the unknown radius, use the given volume:

250\pi = \pi r^2(10)

. Solving this yields

r^2 = 25

, so

r = 5

. Substituting

r = 5

and

h = 10

into the lateral surface area formula gives

2\pi(5)(10) = 100\pi

square feet.

Step-by-Step Solution

1

Set up the volume formula for a cylinder to solve for the radius.

V = \pi r^2 h \Rightarrow 250\pi = \pi r^2 (10)

The volume and height are given, so the radius is the only unknown variable.

2

Solve for the radius (

r

).

250\pi = 10\pi r^2 \Rightarrow r^2 = 25 \Rightarrow r = 5

feet

Dividing both sides by

10\pi

isolates

r^2

, and taking the square root gives the radius.

3

Identify the correct formula for the area to be painted.

Lateral Surface Area =

2\pi r h

The problem specifies painting only the exterior curved side (lateral surface), excluding the top and bottom bases.

4

Calculate the lateral surface area.

2\pi(5)(10) = 100\pi

square feet

Substituting

r = 5

and

h = 10

into the formula yields the final painted area.

Key Concept

Solving for an unknown dimension using the volume formula, then calculating the lateral surface area of a cylinder.

Question 8Question →

A solid rectangular block has a square base with an area of $25$ square inches. If the total volume of the block is $150$ cubic inches, what is the total surface area of the block in square inches?

150

200

170

220

Show answer & explanation

Answer: $170$

Answer

170

To find the total surface area, all dimensions of the block must be known. The side length of the square base is

\sqrt{25} = 5

inches. The height is the volume divided by the base area:

150 \div 25 = 6

inches. The total surface area is the sum of the areas of all six faces: two square bases (

2 \times 25 = 50

) and four identical rectangular sides (

4 \times 5 \times 6 = 120

). The total is

50 + 120 = 170

square inches.

Step-by-Step Solution

1

Find the side length of the square base.

5

inches

The area of a square is the side length squared (

s^2 = 25

), so the side length is the square root of the area.

2

Find the height of the rectangular block.

6

inches

The volume of a rectangular block is the base area multiplied by the height (

V = B \times h

). Dividing the volume (

150

) by the base area (

25

) gives the height.

3

Calculate the total surface area of the block.

170

square inches

The total surface area is the sum of the two square bases (

2 \times 25 = 50

) and the four identical rectangular sides (

4 \times 5 \times 6 = 120

). Adding these together yields

50 + 120 = 170

.

Key Concept

Volume and Surface Area of Rectangular Prisms

Question 9Question →

Pair the given geometric descriptions of 3D figures with their corresponding exact volumes.

Click a left item, then click its matching right item

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Answer

The cylinder matches

80\pi

, the rectangular prism matches

80

, the cube matches

64

, and the sphere matches

36\pi

.

Matching each figure to its correct volume requires applying the specific volume formula for that geometric shape: cylinder (

V = \pi r^2h

), rectangular prism (

V = lwh

), cube (

V = s^3

), and sphere (

V = \frac{4}{3}\pi r^3

).

Step-by-Step Solution

1

Calculate the volume of the cylinder.

V = \pi(4^2)(5) = 80\pi

Applying the cylinder volume formula

V = \pi r^2h

.

2

Calculate the volume of the rectangular prism.

V = 8 \times 5 \times 2 = 80

Applying the rectangular prism volume formula

V = lwh

.

3

Calculate the volume of the cube.

V = 4^3 = 64

Applying the cube volume formula

V = s^3

.

4

Calculate the volume of the sphere.

V = \frac{4}{3}\pi(3^3) = 36\pi

Applying the sphere volume formula

V = \frac{4}{3}\pi r^3

.

Key Concept

Volume and Surface Area of 3D Figures

Question 10Question →

A shipping company uses rectangular boxes that measure $4$ feet in length, $3$ feet in width, and $2$ feet in height. The company needs to completely cover the outside of one box with a protective weather-resistant wrapping. How many square feet of wrapping are required?

14

24

26

52

Show answer & explanation

Answer: 52

Answer

52

The question asks for the amount of wrapping needed to completely cover the outside of the box, requiring the total surface area of a rectangular prism. Using the formula

SA = 2(lw + lh + wh)

and substituting the dimensions

l = 4

,

w = 3

, and

h = 2

yields

SA = 2((4)(3) + (4)(2) + (3)(2)) = 2(12 + 8 + 6) = 2(26) = 52

square feet.

Step-by-Step Solution

1

Identify the required measurement

Calculate the total surface area of the rectangular prism.

The problem asks for the amount of wrapping needed to 'completely cover the outside', which corresponds to surface area rather than volume.

2

Recall the surface area formula for a rectangular prism

SA = 2(lw + lh + wh)

A rectangular box has six faces, composed of three identical pairs of rectangles (top/bottom, front/back, left/right).

3

Substitute the given dimensions and solve

SA = 2((4)(3) + (4)(2) + (3)(2)) = 2(12 + 8 + 6) = 2(26) = 52

square feet.

Applying the measurements

l=4

,

w=3

, and

h=2

evaluates the total combined area of all six faces.

Key Concept

Calculating the total surface area of a rectangular prism.

Question 11Question →

The total surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters?

Show answer & explanation

Answer: 125

Answer

125

The correct answer is derived by first identifying the side length of the cube. Given the total surface area is

150

square centimeters, we divide by

6

to find the area of a single face (

25

square centimeters). The square root of

25

gives a side length of

5

centimeters. Finally, cubing this side length (

5 \times 5 \times 5

) results in a total volume of

125

cubic centimeters.

Step-by-Step Solution

1

Use the total surface area to find the area of a single face of the cube.

150 \div 6 = 25

square centimeters.

A cube has exactly

6

identical square faces. Dividing the total surface area by

6

isolates the area of just one face (

s^2

).

2

Determine the length of one edge of the cube.

s = \sqrt{25} = 5

centimeters.

Because the area of a square face is the side length squared (

s^2

), taking the square root gives the side length.

3

Calculate the total volume of the cube.

V = 5^3 = 125

cubic centimeters.

The formula for the volume of a cube is the side length cubed (

V = s^3

).

Key Concept

Converting between the surface area and volume of a 3D figure by finding the missing dimension.

Estimated Time:45s

Question 12Question →

A solid cylindrical metal rod with a radius of $3$ inches and a length of $8$ inches is melted down and completely recast into a new cylindrical rod. If the new rod must have a radius of $2$ inches and no material is lost, what is its length in inches?

12

72

18

36

Show answer & explanation

Answer: 18

Answer

The length of the new cylindrical rod is 18 inches.

The correct answer is 18. Because no material is lost, the volume of the new rod must strictly equal the volume of the original rod. The original volume is calculated using the formula

V = \pi r^2 h

, resulting in

\pi(3)^2(8) = 72\pi

. Setting this equal to the new rod's volume expression,

72\pi = \pi(2)^2 h

, simplifies the equation to

72\pi = 4\pi h

. Dividing both sides by

4\pi

yields a final length of 18 inches.

Step-by-Step Solution

1

Calculate the volume of the original cylindrical rod.

V_{1} = \pi r^2 h = \pi(3)^2(8) = \pi(9)(8) = 72\pi

cubic inches.

Since the rod is being melted down, the total amount of material (volume) remains constant. We first need the total volume of the original rod.

2

Set up the volume expression for the new cylindrical rod.

V_{2} = \pi r^2 h = \pi(2)^2 h = 4\pi h

.

The new rod is also a cylinder, but with a different radius (2 inches) and an unknown length (height).

3

Equate the two volumes and solve for the unknown length.

72\pi = 4\pi h \rightarrow h = \frac{72\pi}{4\pi} = 18

inches.

Because no material is lost during recasting, the initial volume must equal the final volume.

Key Concept

Conservation of volume in 3D cylinders

Question 13Question →

An architectural firm is analyzing the material requirements and interior spaces of several proposed structures. Match each structural description to its correct calculated capacity (volume) or exterior area (surface area).

Click a left item, then click its matching right item

Show answer & explanation

Answer

The rectangular room matches

960

cubic feet, the cubic pod matches

600

square feet, the cylindrical tank matches

500\pi

cubic feet, and the spherical sculpture matches

400\pi

square feet.

Each structural description correctly aligns with its corresponding formula and numerical result as detailed in the solution steps.

Step-by-Step Solution

1

Calculate the volume of the rectangular room.

V = l \times w \times h = 10 \times 12 \times 8 = 960

cubic feet.

The capacity of a rectangular room is found using the volume formula for a rectangular prism.

2

Calculate the surface area of the cubic storage pod.

SA = 6 \times s^2 = 6 \times (10)^2 = 6 \times 100 = 600

square feet.

A cube has 6 identical square faces, each with an area of

s^2

.

3

Calculate the volume of the cylindrical water tank.

V = \pi \times r^2 \times h = \pi \times (5)^2 \times 20 = 25 \times 20 \times \pi = 500\pi

cubic feet.

The volume of a cylinder is the area of its circular base times its height.

4

Calculate the surface area of the spherical sculpture.

SA = 4 \times \pi \times r^2 = 4 \times \pi \times (10)^2 = 4 \times 100 \times \pi = 400\pi

square feet.

The surface area formula for a sphere is

4

times the area of a circle with the same radius.

Key Concept

Applying appropriate volume and surface area formulas for fundamental 3D figures: rectangular prisms, cubes, cylinders, and spheres.

Question 14Question →

A farmer needs to completely fill a cylindrical grain silo with wheat. If the silo has a diameter of $12$ meters and a height of $10$ meters, what is the maximum capacity of the silo in cubic meters?

60\pi

120\pi

360\pi

1440\pi

Show answer & explanation

Answer: $360\pi$

Answer

The maximum capacity of the silo is

360\pi

cubic meters.

The question asks for the maximum capacity to completely fill the silo, which means finding its volume. The volume of a cylinder is found using the formula

V = \pi r^2 h

. First, the diameter of

12

meters must be halved to find the radius of

6

meters. Substituting the radius and the height (

10

meters) into the formula gives

V = \pi(6)^2(10) = \pi(36)(10) = 360\pi

cubic meters.

Step-by-Step Solution

1

Determine the required measurement and the appropriate formula.

The problem asks for the capacity to 'fill' a cylinder, which requires the volume formula:

V = \pi r^2 h

.

Capacity refers to the interior space (volume) of a 3D figure.

2

Find the radius of the circular base.

The radius is half of the diameter:

12 \div 2 = 6

meters.

The volume formula requires the radius, but the problem provides the diameter.

3

Substitute the values into the formula and solve.

V = \pi(6)^2(10) = \pi(36)(10) = 360\pi

cubic meters.

Executing the calculation according to the order of operations yields the final volume.

Key Concept

Volume of a Cylinder

Estimated Time:1m 0s

Question 15Question →

A community center is repainting the inside of an empty rectangular swimming pool. The pool is $25$ meters long, $10$ meters wide, and has a uniform depth of $2$ meters. If the entire bottom and all four interior walls are to be painted, what is the total area, in square meters, that needs to be painted?

Show answer & explanation

Answer: 390

Answer

390

The correct total area is found by adding the area of the bottom (

250

m

^2

) and the areas of the four vertical walls (

100

m

^2

and

40

m

^2

). Because it is a swimming pool, there is no top to paint, making the total area

390

square meters.

Step-by-Step Solution

1

Determine which surfaces of the rectangular pool need to be painted.

One bottom surface and four side walls (two long and two short). The top is open.

A pool is shaped like an open-top rectangular prism, so we only need to calculate the area of 5 faces instead of 6.

2

Calculate the area of the bottom surface.

Area =

25 \times 10 = 250

square meters.

The bottom is a rectangle bounded by the length and width of the pool.

3

Calculate the combined area of the four side walls.

Long walls =

2 \times (25 \times 2) = 100

square meters. Short walls =

2 \times (10 \times 2) = 40

square meters.

There are two identical walls along the length and two identical walls along the width. The area of each wall is its base dimension multiplied by the depth.

4

Sum the areas of all the painted surfaces.

250 + 100 + 40 = 390

square meters.

The total area to be painted is the sum of the areas of the individual painted faces.

Key Concept

Surface area of an open rectangular prism

Volume and Surface Area of 3D Figures

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