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ACT Math : Geometry

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Example Question #91 : Solid Geometry

What is the surface area of a composite figure of a cone and a sphere, both with a radius of 5 cm, if the height of the cone is 12 cm? Consider an ice cream cone as an example of the composite figure, where half of the sphere is above the edge of the cone.

Possible Answers:

$65\pi\ cm^2$

$50\pi\ cm^2$

$115\pi\ cm^2$

$125\pi\ cm^2$

$165\pi\ cm^2$

Correct answer:

$115\pi\ cm^2$

Explanation:

Calculate the slant height height of the cone using the Pythagorean Theorem. The height will be the height of the cone, the base will be the radius, and the hypotenuse will be the slant height.

s^2=r^2+h^2

$s=\sqrt{169}=13$

The surface area of the cone (excluding the base) is given by the formula $A=\pi rs$ . Plug in our values to solve.

$A_{cone}=\pi(5)(13)=65\pi\ cm^2$

The surface area of a sphere is given by $A=4\pi r^2$ but we only need half of the sphere, so the area of a hemisphere is $A=2\pi r^2$ .

$A_{hemisphere}=2\pi(5)^2=50\pi\ cm^2$

So the total surface area of the composite figure is $65\pi\ cm^2+50\pi\ cm^2=115\pi\ cm^2$ .

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

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

The surface area of a sphere is found using the formula $SA=4\pi r^2$ .

Suppose a sphere has a surface area of $16\pi cm^2$ . What is its volume?

Possible Answers:

$\frac{13}{2}\pi cm^3$

$\frac{32}{3}\pi cm^3$

$\frac{15}{3}\pi cm^3$

$\frac{14}{3}\pi cm^3$

$\frac{16}{3}\pi cm^3$

Correct answer:

$\frac{32}{3}\pi cm^3$

Explanation:

The first step is to use the surface area formula to find the radius of the sphere.

$A=4\pi r^2$

$16\pi=4\pi r^2$

16=4r^2

4=r^2

2=r

The next step is to plug the value of the radius into the volume formula.

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

$V=\frac{4}{3}\pi (2)^3$

$V=\frac{4}{3}\pi(8)$

$V=\frac{32}{3}\pi cm^3$

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

What is the surface area of a sphere with a diameter, in centimeters, $\small \small \small d=6$ ?

The surface area (SA) of a sphere is calculated using the formula $\small \small SA=4\pi r^{2}$ ?

Possible Answers:

$24\pi$ $\small \small \small cm^{2}$

$\ 36\pi$ $\small \small \small cm^{2}$

$144\pi$ $\small \small \small cm^{2}$

$30\pi$ $\small \small \small cm^{2}$

$12\pi$ $\small \small \small cm^{2}$

Correct answer:

$\ 36\pi$ $\small \small \small cm^{2}$

Explanation:

If $\small d=6$ , then $\small r=3$ . Plug the radius into the equation for surface area to get

$\small \small \small \small SA=4\pi (3)^{2}=36\pi$ .

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

What is the surface area of a sphere that has a diameter of eight inches? Reduce any fractions in your answer and leave your answer in terms of $\pi$ .

Possible Answers:

$64 \pi inches^3$

$256 \pi inches^2$

$64\pi inches^2$

$256 \pi inches^3$

$\frac{2048}{3} \pi inches^3$

Correct answer:

$64\pi inches^2$

Explanation:

To find the volume of a sphere area of a sphere plug the radius into the following formula given by :

$SA = 4\pi r^2$ .

To find the radius given the diameter, divide the diameter by 2.

$r=\frac{d}{2}$

$r=\frac{8}{2}=4$
Thus:
$SA = 4\pi 4^2 = 64 \pi inches^2$

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

What is the surface area of a sphere, in inches, that has a surface area equal to its volume? Leave your answer in terms of $\pi$ .

Possible Answers:

$108\pi in^2$

$9\pi in^2$

$36\pi in^3$

$36\pi in^2$

$27\pi in^2$

Correct answer:

$36\pi in^2$

Explanation:

To find the radius of a sphere that has a volume equal to it's surface area, begin by setting the volume and surface area formulas for a sphere equal to each other and solving for the radius:
$\\\frac{4}{3}\pi r^3 = 4\pi r^2 \newline \newline \frac{4}{3}r^3 = 4r^2 \newline \newline 4r^3 = 12r^2 \newline r^3 = 3r^2 \newline r = 3$

Next we plug the answer for our radius into the formula for the surface area:

$4\pi 3^2 = 36\pi in^2$ , remember to check the units.

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

Find the surface area of a sphere whose radius is .

Possible Answers:

$108\pi$

$36\pi$

$12\pi$

$4\pi$

Correct answer:

$36\pi$

Explanation:

To solve, simply remember the following formula:

$SA=4\pi{r^2}=4*\pi*3^2=36\pi$

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

Find the surface area of a sphere whose radius is . Thus,

Possible Answers:

$48\pi$

$32\pi$

$12\pi$

$36\pi$

Correct answer:

$36\pi$

Explanation:

To find surface area, simply use the formula. Thus,

$SA=4\pi{r^2}=4\cdot\pi\cdot3^2=4\cdot\pi\cdot9=36\pi$

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Example Question #4 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere whose side diameter is .

Possible Answers:

$4\pi$

$3\pi$

$2\pi$

$8\pi$

Correct answer:

$4\pi$

Explanation:

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

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

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

For a sphere the volume is given by V = (4/3)πr³ and the surface area is given by A = 4πr². If the sphere has a surface area of 256π, what is the volume?

Possible Answers:

615π

750π

683π

300π

Correct answer:

683π

Explanation:

Given the surface area, we can solve for the radius and then solve for the volume.

4πr² = 256π

4r² = 256

r² = 64

r = 8

Now solve the volume equation, substituting for r:

V = (4/3)π(8)³

V = (4/3)π*512

V = (2048/3)π

V = 683π

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

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate volume of the basketball? Remember that the volume of a sphere is calculated by V=(4πr³)/3

Possible Answers:

434.19 cu.in.

92.48 cu.in.

138.43 cu.in.

8557.46 cu.in.

3468.05 cu.in.

Correct answer:

434.19 cu.in.

Explanation:

To find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get [29.5]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. Then we would plug into the formula for volume V=(4π〖(4.7)〗³) / 3 (The information given of 22 ounces is useless)

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ACT Math : Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #91 : Solid Geometry

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

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

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

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

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

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

Example Question #4 : How To Find The Surface Area Of A Sphere

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

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

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