GRE Math : How to find the diameter of a sphere

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

A cube with a surface area of 216 square units has a side length that is equal to the diameter of a certain sphere. What is the surface area of the sphere?

Possible Answers:

$108\pi$

$24\pi$

$72\pi$

$36\pi$

Correct answer:

$36\pi$

Explanation:

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

SA=6s^2

s= length of one side of the cube

216=6s^2

36=s^2

s=6

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

Diameter=6

$Radius=\frac{Diameter}{2}=3$

$SA=4\pi r^2$

$=4\pi(3)^2$

$=4\pi(9)$

$=36\pi$

The surface area of the sphere is $36\pi$ .

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

The surface area of a sphere is $36\pi$ . What is its diameter?

Possible Answers:

Correct answer:

Explanation:

The surface area of a sphere is defined by the equation:

$SA = 4\pi r^2$

For our data, this means:

$36\pi = 4\pi r^2$

Solving for , we get:

9=r^2 or r=3

The diameter of the sphere is .

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

The volume of one sphere is $432\pi x^3$ . What is the diameter of a sphere of half that volume?

Possible Answers:

$6x\sqrt[3]{12}$

$6x\sqrt[3]{6}$

$3x\sqrt[3]{6}$

$3x\sqrt[3]{12}$

Correct answer:

$6x\sqrt[3]{6}$

Explanation:

Do not assume that the diameter will be half of the diameter of a sphere with volume of $432\pi x^3$ . Instead, begin with the sphere with a volume of $216\pi x^3$ . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

$216 \pi x^3=\frac{4}{3}\pi r^3$

Solving for , we get:

162x^3=r^3

If you take the cube-root of both sides, you have:

$r = \sqrt[3]{162x^3}$

First, you can factor out an :

$r = x\sqrt[3]{162}$

Next, factor the 162 :

$r = x\sqrt[3]{2*3^4}$

Which simplifies to:

$r = 3x\sqrt[3]{6}$

Thus, the diameter is double that or:

$6x\sqrt[3]{6}$

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GRE Math : How to find the diameter of a sphere

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Example Questions

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

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

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

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