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ACT Math : Solid Geometry

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Example Questions

Example Question #14 : Spheres

If the diameter of a sphere is , find the approximate volume of the sphere?

Possible Answers:

$334\pi$

$205\pi$

$167\pi$

$95\pi$

$143\pi$

Correct answer:

$167\pi$

Explanation:

The volume of a sphere = $\frac{4}{3}\pi r^{3}$

Radius is $\frac{1}{2}$ of the diameter so the radius = 5.

$V=\frac{4}{3}\pi5^{3}$

or $V=\frac{500}{3\pi}$

which is approximately $167\pi$

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Example Question #26 : Spheres

What is the volume of a sphere with a diameter of inches? Leave your answer in terms of $\pi$ .

Possible Answers:

$576\pi in^3$

$36\pi in^3$

$288\pi in^3$

$288\pi in^2$

$2404\pi in^3$

Correct answer:

$288\pi in^3$

Explanation:

To find the volume of a sphere we use the sphere volume formula:

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

First we need to find the radius of the sphere. A sphere has a radius of half the diameter. So we see that 12/2 =6=r .

Next we plug 6 in for our radius and get
$V = \frac{4}{3}\pi 6^3=288\pi in^3$

(don't forget your units).

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Example Question #5 : How To Find The Volume Of A Sphere

What is the volume of a sphere with a diameter of $\textup{6 inches}$ (reduce all fractions)?

Possible Answers:

$\frac{864}{3}\pi\textup{ in}^3$

$\frac{108}{3}\pi\textup{ in}^3$

$108\pi\textup{ in}^3$

$36\pi\textup{ in}^3$

$144\pi\textup{ in}^3$

Correct answer:

$\frac{108}{3}\pi\textup{ in}^3$

Explanation:

The formula for the volume of a sphere is:
$v = \frac{4}{3}\pi * r^3$ , thus we need to just determine the radius and plug it into the equation. Remember that 2r = d and so

$6 = 2r \newline 3 = r$

And plugging in we get $\frac{108}{3}\pi \textup{in}^3$

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Example Question #5 : How To Find The Volume Of A Sphere

What is the volume of a sphere with a surface area of $576\pi\textup{ ft}^2$ ? (Simplify all fractions in your answer.)

Possible Answers:

$2304 \pi\textup{ ft}^3$

$2304 \pi\textup{ ft}^2$

$18432\pi\textup{ ft}^3$

$288\pi\textup{ ft}^3$

$2304\pi\textup{ ft}^3$

Correct answer:

$2304\pi\textup{ ft}^3$

Explanation:

First find the radius from the surface area set the given surface area equal to the surface area formula and solve for the radius.
$576\pi = 4\pi r^2$

$144\pi = \pi r^2 \newline 144 = r^2 \newline 12 = r$

Now plug the radius into the volume formula:
$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi 12^3 = 2304 \pi\textup{ ft}^3$

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Example Question #6 : How To Find The Volume Of A Sphere

If Ariana’s orange has twice the radius of Autumn’s orange, the volume of Ariana’s orange is how many times larger than the volume of Autumn’s orange?

Possible Answers:

Correct answer:

Explanation:

Define the radius of Autumn’s orange as r. The volume of her orange is $\pi r^3$ . Ariana’s orange has twice the radius of Autumn’s, so the radius of her orange is , and the volume is $\pi(2r)^3 = 8\pi r^3$ , which is 8 times larger than Autumn’s orange.

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Example Question #111 : Solid Geometry

Find the volume of a sphere whose diameter is .

Possible Answers:

$64\pi$

$256\pi$

$\frac{64}{3}\pi$

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

$\frac{256}{3}\pi$

Correct answer:

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

Explanation:

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

$V=\frac{4}{3}\pi{r^3}=\frac{4}{3}*\pi*2^3=\frac{4}{3}*8*\pi=\frac{32}{3}\pi$

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

A certain cylinder has diameter that is twice the length of its height. If the volume of the cylinder is $64\pi$ cubic inches, what is its radius?

Possible Answers:

$4\pi \ inches$

$4\ inches$

$16\ inches$

$8\ inches$

$8\pi \ inches$

Correct answer:

$4\ inches$

Explanation:

The volume of a cylinder is:

$V=\pi r^{2}h$

You can think of the volume as the area of the base times the height. Since it is given that the diameter is twice the length of the height, the radius (half the diameter) equals the height. If it helps to visualize these dimensions, draw the cylinder described.

The equation can be rewritten, using the height in terms of the radius.

h=r

$V=\pi r^2(r)$

$V=\pi r^{3}$

Plug in the given volume to solve for the radius.

$64\pi=\pi r^{3}$

$64=r^{3}$

$4=r$

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

The radius of a cylinder is five and its height is nine. What is its volume?

Possible Answers:

$225\pi$

$2025\pi$

$25\pi+9$

$45\pi$

$405\pi$

Correct answer:

$225\pi$

Explanation:

To solve this question, you must remember that the formula for volume is the product of the area of the base and the height. The area of the base of this cylinder is $\pi r^2$ .

$V=A_b*h=\pi r^2h$

Plug in the given radius and height to solve.

$V=\pi(5)^2(9)$

$V=\pi(25)(9)$

$V=225\pi$

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Example Question #1 : Cylinders

What is the volume of a round metal washer with an outer radius of 8 in, an inner radius of 2 in, and a thickness of 0.5 in?

Possible Answers:

$6\pi\ in^3$

$10\pi\ in^3$

$24\pi\ in^3$

$30\pi\ in^3$

$16\pi\ in^3$

Correct answer:

$30\pi\ in^3$

Explanation:

The volume of a cylinder is given by the formula: $V=\pi r^2h$ .

For a shape with a hole through the center, the final volume is equal to the total volume of the shape minus the volume of the inner hole. In this question, we are looking for the volume given by the larger radius minus the volume given by the smaller radius. The height is equal to the thickness of the washer.

$V=\pi r_1^2h-\pi r_2^2h$

$V=\pi(8)^2(0.5)-\pi(2)^2(0.5)$

$V=\pi(64)(0.5)-\pi(4)(0.5)$

$V=32\pi-2\pi$

$V=30\pi\ in^3$

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Example Question #4 : How To Find The Volume Of A Cylinder

How much more volume can a cylinder hold than a cone given that both have the same radius and height?

Here B represents the area of the Base, and h the height.

Possible Answers:

None of the answers are correct

$\frac{1}{2}Bh$

$\frac{1}{3}Bh$

$\frac{1}{4}Bh$

$\frac{2}{3}Bh$

Correct answer:

$\frac{2}{3}Bh$

Explanation:

Cylinder: V=Bh Cone: $V=\frac{1}{3}Bh$

Thus the difference is 2/3Bh and that means a cylinder can hold 2/3Bh more given the same radius and height.

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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #14 : Spheres

Example Question #26 : Spheres

Example Question #5 : How To Find The Volume Of A Sphere

Example Question #5 : How To Find The Volume Of A Sphere

Example Question #6 : How To Find The Volume Of A Sphere

Example Question #111 : Solid Geometry

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

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

Example Question #1 : Cylinders

Example Question #4 : How To Find The Volume Of A Cylinder

All ACT Math Resources

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