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SAT Math : How to find the surface area of a sphere

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Example Question #91 : Solid Geometry

Find the surface area of a sphere with radius 1.

Possible Answers:

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

$2\pi$

$4\pi$

$16\pi$

Correct answer:

$4\pi$

Explanation:

To solve, use the formula for the surface are a of a sphere.

Substitute in the radius of one into the following equation.

Thus,

$SA=4\pi{r^2}=4*\pi*1^2=4\pi$

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Example Question #11 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere whose radius is 5.

Possible Answers:

$100\pi$

$25\pi$

$625\pi$

$4\pi$

Correct answer:

$100\pi$

Explanation:

To solve, simply use the formula for the surface area of a sphere. Thus,

$SA=4\pi{r^2}=4*\pi*5^2=4*25*\pi=100\pi$

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Example Question #101 : Solid Geometry

Find the surface area of a sphere with radius 4.

Possible Answers:

$64\pi$

$32\pi$

$20\pi$

$16\pi$

Correct answer:

$64\pi$

Explanation:

To solve, simply use the formula for the surface area of a sphere. Thus,

$SA=4\pi{r^2}=4*\pi*4^2=4*\pi*16=64\pi$

The surface area for a sphere is one of those formulas you are going to have to memorize. There isn't exactly an easy wasy to derive it. My only trick for differentiating it from other circular formulas is the fact that area is two-dimensional, so you only square the r, not cube it.

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Example Question #102 : Solid Geometry

The surface area of a given sphere is $200\pi$ . What is the radius of the sphere?

Possible Answers:

$5\sqrt{2}$

$5\pi$

$10\sqrt{2}$

Correct answer:

$5\sqrt{2}$

Explanation:

The surface area of a given sphere is represented by the equation

$A=4\pi r^2$

Substituting in our given surface area, we can simplify this equation and solve for r.

$200\pi= 4\pi r^2$

r^2=50

$r=\sqrt{50}=5\sqrt{2}$

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Example Question #21 : Spheres

Find the surface area of a sphere whose radius is $\frac{\pi}{2}$ .

Possible Answers:

$\pi^2$

$4\pi^2$

$\pi^3$

$\frac{\pi}{4}$

Correct answer:

$\pi^3$

Explanation:

The equation for the surface area of a sphere is $A = 4 \pi r^2$ where represents the sphere's radius.

With our radius-value, we find:

$A = 4 \pi r^2 = 4\pi (\frac{\pi}{2})^2 = 4\pi \cdot \frac{\pi^2}{4} = \pi^3$

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Example Question #22 : Spheres

Give the area of the largest circle that can be drawn entirely on the surface of a sphere with surface area 720.

Possible Answers:

$\frac{360 }{\pi}$

180

$\frac{180}{\pi}$

$\frac{720}{\pi}$

360

Correct answer:

180

Explanation:

The circle of greatest size that can possibly be constructed on a sphere will have the same radius as the sphere as can be seen in the diagram below.

Sphere

This circle has area $A = \pi r^{2}$

The surface area of a sphere is related to its radius by the formula

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

Since the surface area is 720,

$4 \pi r^{2} = 720$

If both sides are divided by 4, it can be seen that

$\frac{4 \pi r^{2} }{4}= \frac{720}{4}$

$\pi r^{2} = 180$

A =180

making this the area of the circle.

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Example Question #23 : Spheres

Let be a point on a sphere, and be the point on the sphere farthest from . The shortest distance from to along the surface is $32 \pi$ . Give the surface area of the sphere.

Possible Answers:

$1,024 \pi$

$2,048 \pi$

$16,384 \pi$

$4,096 \pi$

$8,192 \pi$

Correct answer:

$4,096 \pi$

Explanation:

The diagram below shows the sphere with the points in question as well as the curve that connects them.

Sphere

The curve connecting them is a semicircle whose radius coincides with that of the sphere. Given radius , a semicircle has length

$l = \frac{1}{2} C= \frac{1}{2} \cdot 2 \pi r = \pi r$

Setting $l = 32\pi$ and solving for :

$\pi r = 32 \pi$

$\frac{\pi r }{\pi}= \frac{32\pi}{\pi}$

r = 32 .

The surface area of a sphere, given its radius , is equal to

$A= 4 \pi r ^{2}$ .

Setting r = 32 :

$A= 4 \pi \cdot 32 ^{2}= 4 \cdot 32 ^{2} \cdot \pi = 4 \cdot 1,024\cdot \pi = 4,096 \pi$

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SAT Math : How to find the surface area of a sphere

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #91 : Solid Geometry

Example Question #11 : How To Find The Surface Area Of A Sphere

Example Question #101 : Solid Geometry

Example Question #102 : Solid Geometry

Example Question #21 : Spheres

Example Question #22 : Spheres

Example Question #23 : Spheres

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