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SAT Math : How to find the volume of a cube

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Example Questions

Example Question #731 : Geometry

A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere. What is the volume of the cube?

Possible Answers:

$\frac{8\sqrt{3}}{9}$

$\frac{10\sqrt{3}}{7}$

$\frac{7\sqrt{3}}{10}$

$\frac{9\sqrt{3}}{8}$

Correct answer:

$\frac{8\sqrt{3}}{9}$

Explanation:

Cube

To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points A, B, C are midpoints of the edges of the inscribed cube. Therefore point , a vertex of the smaller cube, is also the center of the sphere. Point lies on the circumference of the sphere, so ED = 1 . is also the hypotenuse of right triangle FED . Similarly, is the hypotenuse of right triangle FBD . If we let BD = x , then, by the properties of a right triangle, $FD = x\sqrt2$ .

Using the Pythagorean Theorem, we can now solve for :

FD^2 + FE^2 = DE^2

$(x\sqrt{2})^2 + x^2 = 1^2$

2x^2 + x^2 = 1

3x^2 = 1

$x^2 = \frac{1}{3}$

$x=\sqrt{\frac{1}{3}}$

Since the side of the inscribed cube is , the volume is $V= (2x)^3=8x^3=8\sqrt{\frac{1}{3}}^3=\frac{8}{3\sqrt{3}}=\frac{8}{3\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{8\sqrt{3}}{9}$ .

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Example Question #732 : Geometry

A perfect cube has a volume of 8 cubic centimeters. If the height, length and width of the cube were doubled, what would be the volume of the cube?

Possible Answers:

$128\:cm^3$

$16\:cm^3$

$64\:cm^3$

$24\:cm^3$

$48\:cm^3$

Correct answer:

$64\:cm^3$

Explanation:

Volume is calculated by height x width x length: $V=h\cdot w \cdot l$

For a cube, the height, width, and length are all the same value, so the equation can be simplified to V=s^3 , where is the length of one edge of the cube.

We know that for the initial cube, $V=8\:cm^3$ , so we can substitute this into the volume equation and solve for the length of one of the cube's sides:

V=s^3

8=s^3

$\sqrt[3]8=\sqrt[3]{s^3}$

2=s

So, one edge of the initial cube is $2\:cm$ long. When doubled, the cube will have edges that are each $4\:cm$ long. We can solve for the final volume of the cube by substituting $4\:cm$ into the equation for the volume of a cube and solving:

V=s^3

V=4^3

$V=4\cdot 4\cdot 4=64\:cm^3$

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Example Question #733 : Geometry

What's the volume of a cube with a length of 10a ?

Possible Answers:

100a^3

300a^3

30a^3

3600a^2

1000a^3

Correct answer:

1000a^3

Explanation:

Write the volume formula for a cube.

V=s^3

Substitute the side.

V=s^3=(10a)^3 = 1000a^3

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Example Question #32 : Cubes

If the surface area of a cube is 30, what is the volume of the cube?

Possible Answers:

$5\sqrt5$

$\sqrt5$

$5\sqrt3$

125

900

Correct answer:

$5\sqrt5$

Explanation:

Write the surface area formula of the cube.

A=6s^2

Substitute the surface area and find the side.

30=6s^2

s^2=5

$s=\sqrt5$

Write the volume for the cube.

V=s^3

Substitute the side to the volume formula.

$V=s^3=5\sqrt5$

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Example Question #735 : Geometry

The volumes of six cubes form an arithmetic sequence. The two smallest cubes have sidelengths 10 and 12, respectively. Give the volume of the largest cube.

Possible Answers:

4,640

1,920

5,368

8,000

2,400

Correct answer:

4,640

Explanation:

The volume of a cube is equal to the length of a side raised to the third power. The two smallest cubes will have volumes:

$V_{1} = 10^{3} = 1,000$

and

$V_{2} = 12^{3} = 1,728$ ,

respectively.

The volumes form an arithmetic sequence with these two volumes as the first two terms, so their common difference is

$d = V_{2} - V_{1} = 1,728 - 1,000 = 728$ .

The volume of the largest, or sixth-smallest, cube, is

$V_{6} = (6-1)d + V_{1}$

=5 (728) + 1,000

=3,640 + 1,000

= 4,640

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Example Question #734 : Geometry

Find the volume of a cube with side length 4.

Possible Answers:

254

Correct answer:

Explanation:

To solve, simply use the formula for the volume of a cube.

Substitute in the side length of four into the following equation.

Thus,

V=s^3=4^3=64

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Example Question #735 : Geometry

Find the volume of a cube given side length is 1.

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for the volume of a cube. Thus,

V=s^3=1^3=1

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Example Question #17 : How To Find The Volume Of A Cube

A cube has a surface area of 150 . What is its volume?

Possible Answers:

150

125

100

Correct answer:

125

Explanation:

Remember that a cube's surface area, because it's comprised of six identical squares, can be stated as 6s^2 . With that in mind,

6s^2 = 150

s^2 = 25

$s = \sqrt{25} = 5$

The last step is easy: V = s^3 = 5^3 = 125

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Example Question #16 : How To Find The Volume Of A Cube

What is the volume of a cube with a side length of 7.5 cm?

(Round two the nearest two places)

Possible Answers:

274.63cm^2

421.88cm^3

512cm^2

857.38cm^3

Correct answer:

421.88cm^3

Explanation:

The formula for volume of a cube is,

V=s^3

where

s=side .

The side length of the cube is given as 7.5cm.

s=7.5

Substituting this into the formula for a cube's volume is as follows.

$\\V=(7.5)^3 \\V=(7.5)(7.5)(7.5) \\V=421.875\approx 421.88cm^3$

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Example Question #741 : Sat Mathematics

Find the volume of a cube whose side length is 7cm.

Possible Answers:

7cm^3

196cm^3

343cm^3

49cm^3

21cm^3

Correct answer:

343cm^3

Explanation:

The volume of a cube is length*width*height. In a cube all the side lengths are equal. Volume=7cm*7cm*7cm=343cm^3

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SAT Math : How to find the volume of a cube

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #731 : Geometry

Example Question #732 : Geometry

Example Question #733 : Geometry

Example Question #32 : Cubes

Example Question #735 : Geometry

Example Question #734 : Geometry

Example Question #735 : Geometry

Example Question #17 : How To Find The Volume Of A Cube

Example Question #16 : How To Find The Volume Of A Cube

Example Question #741 : Sat Mathematics

All SAT Math Resources

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