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SAT Math : Geometry

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Example Questions

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

If a cube has its edges increased by a factor of 5, what is the ratio of the new volume to the old volume?

Possible Answers:

100:1

15:1

5:1

125:1

25:1

Correct answer:

125:1

Explanation:

A cubic volume is side*side*side . Let the original sides be 1, so that the original volume is 1. Then find the volume if the sides measure 5. This new volume is 125. Therefore, the ratio of new volume to old volume is 125: 1.

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Example Question #51 : Solid 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:

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

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

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

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

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 #1391 : Concepts

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:

$16\:cm^3$

$64\:cm^3$

$24\:cm^3$

$48\:cm^3$

$128\:cm^3$

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 #731 : Geometry

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

Possible Answers:

300a^3

1000a^3

3600a^2

30a^3

100a^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 #732 : 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 #733 : 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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SAT Math : Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

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

Example Question #51 : Solid Geometry

Example Question #1391 : Concepts

Example Question #731 : Geometry

Example Question #32 : Cubes

Example Question #735 : Geometry

Example Question #732 : Geometry

Example Question #733 : Geometry

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

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

All SAT Math Resources

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