GRE Math : Solid Geometry

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Example Questions

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

What is the volume of a sphere with a radius of 3?

Possible Answers:

24π

36π

112π

48π

96π

Correct answer:

36π

Explanation:

Volume of a sphere = 4/3 * πr³ = 4/3 * π * 3³= 36π

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Example Question #344 : Geometry

How many times greater is the volume of a sphere with radius of 3 than the volume of a sphere with radius of $\small \sqrt{3}$ ?

Possible Answers:

$\small 3$

$\small \sqrt{3}$

$\small 3\sqrt{3}$

$\small 9\sqrt{3}$

$\small 9$

Correct answer:

$\small 3\sqrt{3}$

Explanation:

The formula for the volume of a sphere is $\small \small V = \frac{4}{3} \pi r^{3}$ .

Use this to find the volume of each sphere, then take the quotient of the two volumes to determine the relationship between the two.

For the first sphere, $\small V = \frac{4}{3}\pi 3^{3} = \frac{4}{3} \pi * 27$ .

For the second sphere, $\small \small V = \frac{4}{3} \pi * (\sqrt{3})^{3} = \frac{4}{3} \pi * 3\sqrt{3}$ .

Since both volumes have factors of 4/3 and pi, we'll ignore those and divide by the remaining factors. Keep in mind that the volume of the first sphere must be in the numerator due to the wording of the question.

$\small \frac{27}{3\sqrt{3}} = \frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} =3\sqrt{3}$

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Example Question #1 : How To Find The Radius Of A Sphere

If a sphere has a volume of 268.08 cubic inches, what is the approximate radius of the sphere?

Possible Answers:

4 in

64 in

4.5 in

8 in

5.9 in

Correct answer:

4 in

Explanation:

The formula for the volume of a sphere is

$v=\frac{4}{3}(\pi r^3)$ where is the radius of the sphere.

Therefore,

$r=\sqrt[3]{\frac{3v}{(4\pi)}}$

$r=\sqrt[3]{\frac{3(268.08)}{(4\pi)}}=\sqrt[3]{63.999}=3.999$ , giving us $r\approx4$ .

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Example Question #2 : How To Find The Radius Of A Sphere

A rectangular prism has the dimensions $9\textup{ x }6\textup{ x }12$ . What is the volume of the largest possible sphere that could fit within this solid?

Possible Answers:

$288\pi$

$972\pi$

$2304\pi$

$36\pi$

Correct answer:

$36\pi$

Explanation:

For a sphere to fit into the rectangular prism, its dimensions are constrained by the prism's smallest side, which forms its diameter. Therefore, the largest sphere will have a diameter of , and a radius of .

The volume of a sphere is given as:

$\frac{4}{3}\pi r^3$

And thus the volume of the largest possible sphere to fit into this prism is

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

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Example Question #1 : How To Find The Radius Of A Sphere

What is the radius of a sphere with volume $36\pi$ cubed units?

Possible Answers:

Correct answer:

Explanation:

The volume of a sphere is represented by the equation $\frac{4}{3}\pi r^3$ . Set this equation equal to the volume given and solve for r:

$36\pi=\frac{4}{3}\pi r^3$

$36\pi\cdot \frac{3}{4}=\pi r^3$

$27\pi=\pi r^3$

27=r^3

r=3

Therefore, the radius of the sphere is 3.

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GRE Math : Solid Geometry

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

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

Example Question #344 : Geometry

Example Question #1 : How To Find The Radius Of A Sphere

Example Question #2 : How To Find The Radius Of A Sphere

Example Question #1 : How To Find The Radius Of A Sphere

All GRE Math Resources

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