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SAT Math : Geometry

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Example Questions

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

What is the surface area of a hemisphere with a diameter of $4\ cm$ ?

Possible Answers:

$20\pi \ cm^{2}$

$16\pi \ cm^{2}$

$12\pi \ cm^{2}$

$8\pi \ cm^{2}$

$18\pi \ cm^{2}$

Correct answer:

$12\pi \ cm^{2}$

Explanation:

A hemisphere is half of a sphere. The surface area is broken into two parts: the spherical part and the circular base.

The surface area of a sphere is given by $SA = 4\pi r^{2}$ .

So the surface area of the spherical part of a hemisphere is $SA = 2\pi r^{2}$ .

The area of the circular base is given by $A = \pi r^{2}$ . The radius to use is half the diameter, or 2 cm.

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Example Question #9 : How To Find The Surface Area Of A Sphere

Six spheres have surface areas that form an arithmetic sequence. The two smallest spheres have radii 4 and 6. Give the surface area of the largest sphere.

Possible Answers:

$464 \pi$

$\frac{3,296}{3} \pi$

$784 \pi$

$\frac{10, 976 }{3}\pi$

$196 \pi$

Correct answer:

$464 \pi$

Explanation:

The surface area of a sphere with radius can be determined using the formula

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

The smallest sphere, with radius r = 4 , has surface area

$S_{1} = 4 \pi \cdot 4^{2} = 4 \pi \cdot 16 = 64 \pi$

The second-smallest sphere, with radius r = 6 , has surface area

$S_{2} = 4 \pi \cdot 6^{2} = 4 \pi \cdot 36 = 144 \pi$

The surface areas are in an arithmetic sequence; their common difference is the difference of these two surface areas, or

$d = S_{2} - S_{1}= 144 \pi -64 \pi = 80 \pi$

Since the six surface areas are in an arithmetic sequence, the surface area of the largest of the six spheres - that is, the sixth-smallest sphere - is

$S_{6}= S_{1}+ 5d$

$= 64 \pi + 5 \cdot 80 \pi$

$= 64 \pi + 400 \pi$

$= 464 \pi$

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

The radii of six spheres form an arithmetic sequence. The smallest and largest spheres have radii 10 and 30, respectively. Give the surface area of the second-smallest sphere.

Possible Answers:

$784 \pi$

$\frac{10,976 }{3} \pi$

$\frac{6,400 }{9} \pi$

None of the other responses gives a correct answer.

$\frac{1,024,000}{81} \pi$

Correct answer:

$784 \pi$

Explanation:

The radii of the spheres form an arithmetic sequence, with

$r_{1} =1 0$ and $r_{6} =30$

The common difference can be computed as follows:

$r_{6} = (6-1) d + r_{1}$

30=5d + 10

5d= 20

d = 4

The second-smallest sphere has radius

$r_{2} = r_{1}+ d = 10 + 4 = 14$

The surface area of a sphere with radius can be determined using the formula

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

Setting r = 14 , we get

$S = 4 \pi \cdot 14^{2} = 4 \pi 196 = 784 \pi$ .

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Example Question #791 : Geometry

Find the surface area of a sphere with radius 1.

Possible Answers:

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

$16\pi$

$4\pi$

$2\pi$

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 #16 : Spheres

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

Possible Answers:

$100\pi$

$4\pi$

$625\pi$

$25\pi$

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 #792 : Geometry

Find the surface area of a sphere with radius 4.

Possible Answers:

$20\pi$

$64\pi$

$16\pi$

$32\pi$

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 #795 : Geometry

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

Possible Answers:

$5\pi$

$10\sqrt{2}$

$5\sqrt{2}$

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:

$\pi^2$

$\frac{\pi}{4}$

$4\pi^2$

$\pi^3$

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 #21 : 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:

180

360

$\frac{180}{\pi}$

$\frac{720}{\pi}$

$\frac{360 }{\pi}$

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 #22 : 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:

$16,384 \pi$

$2,048 \pi$

$1,024 \pi$

$4,096 \pi$

$8,192 \pi$

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 : Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

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

Example Question #9 : How To Find The Surface Area Of A Sphere

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

Example Question #791 : Geometry

Example Question #16 : Spheres

Example Question #792 : Geometry

Example Question #795 : Geometry

Example Question #21 : Spheres

Example Question #21 : Spheres

Example Question #22 : Spheres

All SAT Math Resources

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