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ACT Math : Solid Geometry

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Example Questions

Example Question #2 : How To Find The Volume Of A Figure

Chemicals to clean a swimming pool cost $0.24 per cubic foot of water. If a pool is 6 feet deep, 14 feet long and 8 feet wide, how much does it cost to clean the pool? Round to the nearest dollar.

Possible Answers:

$\$150$

$\$161$

$\$202$

$\$672$

$\$184$

Correct answer:

$\$161$

Explanation:

The volume of the pool can be determined by multiplying the length, width, and height together.

$V = 6 \times 8 \times 14 = 672 ft^{3}$

Each cubit foot costs 24 cents, so:

$672ft^3\times \$0.24=\$161.28\approx \$161$

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Example Question #1 : How To Find The Volume Of A Cube

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Possible Answers:

Correct answer:

Explanation:

If we let represent the length of an edge on the smaller cube, its volume is $x^{3}$ .

The larger cube has edges three times as long, so the length can be represented as . The volume is $(3x)^{3}$ , which is $27x^{3}$ .

The large cube's volume of $27x^{3}$ is 27 times as large as the small cube's volume of $x^{3}$ .

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Example Question #41 : Cubes

A tank measuring 3in wide by 5in deep is 10in tall. If there are two cubes with 2in sides in the tank, how much water is needed to fill it?

Possible Answers:

134 in^3

150 in^3

16 in^3

8 in^3

160 in^3

Correct answer:

134 in^3

Explanation:

$Volume_{Tank} = Width \cdot Depth \cdot Height$

$Volume_{Tank} = 3in \cdot 5in \cdot 10in = 150 in^3$

$Volume_{Cube} = (Side)^3 = (2in)^3 = 8 in^3$

$Volume_{Water}=Volume_{Tank}-2(Volume_{Cube})$

$Volume_{Water}=150-2(8)$

$Voume_{Water}=150-16$

$Volume_{Water}=134in^3$

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Example Question #3 : How To Find The Volume Of A Cube

What is the volume of a cube that has a side length of inches?

Possible Answers:

63in^3

343in^2

21in^3

49in^2

343 in^3

Correct answer:

343 in^3

Explanation:

We are given the side length of a cube so we simply plug that into the formula for the volume of a cube.

That formula is V =s^3 , and so the correct answer is

V=7^3=343in^3 .

Make sure to check your units, 343in^2 is the correct number, but the units are squared rather than cubed.

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Example Question #3 : How To Find The Volume Of A Cube

What is the volume of a cube with a side of length 1 cm?

Possible Answers:

6cm^3

1cm

3cm^3

0cm^3

1cm^3

Correct answer:

1cm^3

Explanation:

The formula for the volume of a cube with a side of length is:
$\\V = s^3 \\V= 1^3 \\V= 1cm^3$

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Example Question #8 : How To Find The Volume Of A Cube

A cube of sponge has volume 1000 cm^3 . When water is added, the sponge triples in length along each dimension. What is the new volume of the cube, in cubic centimeters?

Possible Answers:

2700cm^3

9000cm^3

30000cm^3

27000 cm^3

3000cm^3

Correct answer:

27000 cm^3

Explanation:

If our original cube has a volume of 1000cm^3 , then the length of one of its edges is $\sqrt[3]{1000} = 10cm$ . Triple each edge to 30cm , then cube the result, and we obtain 30^3 = 27000cm^3

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Example Question #1 : How To Find The Volume Of A Cube

Find the volume of a cube with side length 10.

Possible Answers:

$10\sqrt{2}$

1000

100

600

Correct answer:

1000

Explanation:

To find volume, simply cube the side length. Thus,

V=s^3=10^3=1000

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Example Question #41 : Solid Geometry

Find the volume of a cube whose side length is 0.1.

Possible Answers:

0.1

0.001

0.01

Correct answer:

0.001

Explanation:

To solve, simply use the formula for volume of a cube. The volume of a cube is length times width times height. Since the height, width, and length are equal on a cube the formula becomes side cubed.

We are given the side length of 0.1 thus,

V=s^3=(0.1)^3=0.001

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Example Question #42 : Solid Geometry

Find the volume of a cube given side length is .

Possible Answers:

1000

100

Correct answer:

1000

Explanation:

To solve, simply cube the side length. Thus,

V=s^3=10*10*10=1000

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Example Question #1 : Pyramids

A square pyramid has a volume of 363cm^3 and a height of $\small 9cm$ . What is the perimeter of the base of the pyramid?

Possible Answers:

36cm

44cm

80cm

100cm

121cm

Correct answer:

44cm

Explanation:

The formula for the volume of a pyramid is:

$\small V=\frac{lwh}{3}$

We know that V=363 and h=9 ; however, this still leaves us with two variables in the equation: and . By definition, a square pyramid's base has sides of equal length, meaning that and are the same. Therefore, we can substitute $\small lw$ for $\small l\cdot l$ , or $\small l^2$ .

This gives us a new equation of:

$\small V=\frac{l^2h}{3}$

We then plug in the variables we know:

$\small 363=\frac{9l^2}{3}$

Multiply both sides by 3:

$\small 1089=9l^2$

Divide both sides by 9:

l^2=121

l=11

Therefore, we now know that the length of the pyramid's base is $\small 11cm$ . The question, however, asks for the perimeter of the pyramid's base. Since all of the sides of the base are the same, they must all be $\small 11cm$ . So we multiply $11cm\cdot 4=44cm$ . Therefore, the perimeter of the pyramid's base is $\small 44cm$ .

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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2 : How To Find The Volume Of A Figure

Example Question #1 : How To Find The Volume Of A Cube

Example Question #41 : Cubes

Example Question #3 : How To Find The Volume Of A Cube

Example Question #3 : How To Find The Volume Of A Cube

Example Question #8 : How To Find The Volume Of A Cube

Example Question #1 : How To Find The Volume Of A Cube

Example Question #41 : Solid Geometry

Example Question #42 : Solid Geometry

Example Question #1 : Pyramids

All ACT Math Resources

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