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ACT Math : Solid Geometry

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Example Questions

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

What is the radius of a sphere with a volume of $36\prod \textup{ units}^{3}$ ?

Possible Answers:

$27\textup{ units}$

$12\textup{ units}$

$9\textup{ units}$

$6\textup{ units}$

$3\textup{ units}$

Correct answer:

$3\textup{ units}$

Explanation:

Given the volume of the sphere, $36\prod \textup{units}^{3}$ , you need to use the formula for volume of a sphere $\left ( \left ( \frac{4}{3} \right )\cdot \prod \cdot r^{3}\right )$ and work backwards to find the radius. I would multiply both sides by $\frac{3}{4}$ to get rid of the $\frac{4}{3}$ in the formula. You then have $27\prod=\prod r^{3}$ . Next, divide both sides by $\prod$ so that all vyou have left is $27=r^{3}$ . Finally take the cube root of , to get units for the radius.

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

A cube with sides of 25m is circumscribed by a sphere, such that all eight vertices of the cube are tangent to the sphere. What is the sphere's radius?

Possible Answers:

$\frac{125\sqrt5}{2}$

125

$125\sqrt5$

$\frac{25\sqrt{3}}{2}$

$25\sqrt3$

Correct answer:

$\frac{25\sqrt{3}}{2}$

Explanation:

Solving this problem requires recognizing that since the cube is circumscribed by the sphere, both solids share the same center. Now it is just a matter of finding the diagonal of the cube, which will double as the diameter of the sphere (by definition, any straight line which passes through the center of the sphere). The formula for the diagonal of a cube is $D = s\sqrt{3}$ , where is the length of the side of a cube. (This occurs because you must use the Pythagorean theorem once for each 2-dimensional "corner" you travel to find the diagonal for a 3-dimensional shape, but for the ACT it's much faster to memorize the formula.)

In this case:

$D = s\sqrt3 = 25\sqrt3$

Since the radius is half the diameter, divide the result in half:

$r = \frac{D}{2} = \frac{25\sqrt{3}}{2}$

$r = \frac{25\sqrt{3}}{2}$

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Example Question #4 : Spheres

If a sphere has a volume of $36\pi$ , what is its diameter?

Possible Answers:

$2\sqrt{3}$

$\sqrt{3}$

Correct answer:

Explanation:

1. Use the volume to find the radius:

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

$36\pi = \frac{4}{3}\pi r^{3}$

$\frac{3}{4}(36\pi) = (\frac{4}{3}\pi r^{3})\frac{3}{4}$

$27=r^{3}$

r=3

2. Use the radius to find the diameter:

$d=2r=2\cdot 3=6$

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Example Question #91 : Solid Geometry

A sphere has a volume of $36\pi$ . What is its diameter?

Possible Answers:

Cannot be determined from the information given

144

$3\pi$

Correct answer:

Explanation:

This question relies on knowledge of the formula for volume of a sphere, which is as follows: $V=\frac{4}{3}\pi r^{3}}$

In this equation, we have two variables, and . Additionally, we know that $V=36\pi$ and is unknown. You can begin by rearranging the volume equation so it is solved for , then plug in and solve for :

Rearranged form:

$r=\sqrt[3]{\frac{3}{4\pi}V}$

Plug in $36\pi$ for V

$r=\sqrt[3]{\frac{3}{4\pi}36\pi}$

Simplify the part under the cubed root

1) Cancel the $\pi$ 's since they are in the numerator and denominator.

2) Simplify the fraction and the :

$36*\frac{3}{4}=27$

Thus we are left with

$r=\sqrt[3]{27}$

Then, either use your calculator and enter $27^{\frac{1}{3}}$ Or recall that $3^{3}=27$ in order to find that r=3 .

We're almost there, but we need to go a step further. Dodge the trap answer "" and carry on. Read the question carefully to see that we need the diameter, not the radius.

d=2r

d=2*3=6

is our final answer.

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

A spherical plastic ball has a diameter of $4\:in$ . What is the volume of the ball to the nearest cubic inch?

Possible Answers:

$34\:in^3$

$44\:in^3$

$269\:in^3$

$22\:in^3$

$56\:in^3$

Correct answer:

$34\:in^3$

Explanation:

To answer this question, we must calculate the volume of the ball using the equation for the volume of a sphere. The equation for the volume of a sphere is four-thirds multiplied by pi, which is then multiplied by the radius cubed. The equation can be written like this:

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

We are given the diameter of the sphere in the problem, which is $4\:in$ . To get the radius from the diameter, we divide the diameter by . So, for this data:

$radius=\frac{diameter}{2}=\frac{4}{2}=2$

We can then plug our newly found radius of two into the equation to find the volume. For this data:

$Volume = \frac{4}{3}\pi r^{3}=\frac{4}{3}\pi \cdot (2)^{3} = \frac{4}{3}\pi\cdot 8$

We then multiply $\frac{4}{3}$ by .

$\frac{4}{3}\pi\cdot 8=\frac{32}{3}\pi$

We finally substitute 3.14 for pi and multiply again to get our answer.

$\frac{32}{3}\pi = 33.5$

The question asked us to round to the nearest whole cubic inch. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

$33.5\rightarrow34$

Therefore our answer is $34\:in^3$ .

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Example Question #7 : Spheres

A boulder breaks free on a slope and rolls downhill. It rolls for 355 complete revolutions before grinding to a halt. If the boulder has a volume of 1436 cubic feet, how far in feet did the boulder roll? (Assume the boulder doesn't lose mass to friction). Round $\pi$ to 3 significant digits. Round your final answer to the nearest integer.

Possible Answers:

15606ft

23280ft

9978ft

11747ft

11682ft

Correct answer:

15606ft

Explanation:

The formula for the volume of a sphere is:

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

To figure out how far the sphere rolled, we need to know the circumference, so we must first figure out radius. Solve the formula for volume in terms of radius:

$1436 = \frac{4}{3}\pi r^3$

$\frac{1077}{\pi} =r^3$

$r \approx 6.999932$

Since the answer asks us to round to the nearest integer, we are safe to round to at this point.

To find circumference, we now apply our circumference formula:

$C = 2r\pi = 14\pi$

If our boulder rolled 355 times, it covered that many times its own circumference.

$355\cdot 14\pi = 4970\pi \approx 15606$

Thus, our boulder rolled for 15606ft

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Example Question #91 : Solid Geometry

Find the diameter of a sphere whose radius is .

Possible Answers:

2d^2

4d^2

Correct answer:

Explanation:

To solve, simply remember that diameter is twice the radius. Don't be fooled when the radius is an algebraic expression and incorporates the arbitrary constant . Thus,

$\textup{diameter}=2r=2*2d=4d$

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Example Question #731 : 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 #1001 : Act Math

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:

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

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

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

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

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

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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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #4 : Spheres

Example Question #91 : Solid Geometry

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

Example Question #7 : Spheres

Example Question #91 : Solid Geometry

Example Question #731 : Geometry

Example Question #1001 : Act Math

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

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