How to find the radius of a sphere - SAT Math

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Question

A cube with volume 27 cubic inches is inscribed inside a sphere such that each vertex of the cube touches the sphere. What is the radius, in inches, of the sphere?

Answer

We know that the cube has a volume of 27 cubic inches, so each side of the cube must be ∛27=3 inches. Since the cube is inscribed inside the sphere, the diameter of the sphere is the diagonal length of the cube, so the radius of the sphere is half of the diagonal length of the cube. To find the diagonal length of the cube, we use the distance formula d=√(32+32+32 )=√(3*32 )=3√3, and then divide the result by 2 to find the radius of the sphere, (3√3)/2.

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Question

The surface area of a sphere is 100π square feet. What is the radius in feet?

Answer

S = 4π(r2)

100π = 4π(r2)

100 = 4r2

25 = r2

5 = r

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Question

What is the radius of a sphere with a surface area of 16?

Answer

In order to find the radius, first write the surface area formula for a sphere and substitute the surface area.

$A=4\pi r^2$

$16=4\pi r^2$

Divide by $4\pi$ on both sides in order to isolate the term.

$\frac{16}{4\pi}=\frac{4\pi r^2}{4\pi}$

Simplify both sides of the equation.

$r^2 = \frac{4}{\pi}$

Square root both sides.

$r= \sqrt {\frac{4}{\pi}} = \frac{\sqrt4}{\sqrt{\pi}} = \frac{2}{\sqrt{\pi}}$

The radius is $\frac{2}{\sqrt{\pi}}$ .

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