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ACT Math : Plane Geometry

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Example Questions

Example Question #4 : How To Find The Length Of The Diameter

A circle has an area of $39\pi$ . What is the diameter of the circle?

Possible Answers:

$2\sqrt{39\pi }$

$\sqrt{39}$

$2\sqrt{39}\pi$

$2\sqrt{39}$

$\sqrt{39\pi }$

Correct answer:

$2\sqrt{39}$

Explanation:

The equation for the area of a circle is $\pi r^2$ , which in this case equals $39\pi$ . Therefore, r^2 = 39. The only thing squared that equals an integer (which is not a perfect root) is that number under a square root. Therefore, $r=\sqrt{39}$ . Since diameter is twice the radius, $d=2\sqrt{39}.$

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Example Question #1 : Diameter

Find the diameter given the radius is .

Possible Answers:

$\frac{3}{2}$

Correct answer:

Explanation:

Diameter is simply twice the radius. Therefore, 2*3=6 .

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Example Question #2 : Diameter

Find the length of the diameter of a circle given the area is $4\pi$ .

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for the area of a circle, solve for r, and multiply by 2 to get the diameter. Thus,

$A=\pi{r^2}\Rightarrow r=\sqrt{\frac{A}{\pi}}$

$r=\sqrt{\frac{4\pi}{\pi}}=\sqrt{4}=2$

d=2r=2*2=4

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Example Question #6 : How To Find The Length Of The Diameter

In a group of students, it was decided that a pizza would be divided according to its crust size. Every student wanted 3 inches of crust (measured from the outermost point of the pizza). If the pizza in question had a diameter of 14 inches, what percentage of the pizza was wasted by this manner of cutting the pizza? Round to the nearest hundredth.

Possible Answers:

$66.08\%$

$4.51\%$

$14.66\%$

$5.71\%$

$2.81\%$

Correct answer:

$4.51\%$

Explanation:

What we are looking at is a way of dividing the pizza according to arc lengths of the crust. Thus, we need to know the total circumference first. Since the diameter is , we know that the circumference is $14\pi$ . Now, we want to ask how many ways we can divide up the pizza into pieces of inch crust. This is:

$\frac{14\pi}{3}$ or approximately 14.66... pieces.

What you need to do is take this amount and subtract off . This is the amount of crust that is wasted. You can then merely divide it by the original amount of divisions:

$\frac{0.66076571675237...}{14.66076571675237....}=0.04507034144863....$

(You do not need to work in exact area or length. These relative values work fine.)

This is about $4.51\%$ of the pizza that is wasted.

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Example Question #2 : How To Find The Length Of The Diameter

A circle has an area of $\textup{in}^{2}$ . What is its diameter?

Possible Answers:

5.62 $\textup{in}$

2.81 $\textup{in}$

8.14 $\textup{in}$

7.57 $\textup{in}$

3.78 $\textup{in}$

Correct answer:

7.57 $\textup{in}$

Explanation:

To solve a question like this, first remember that the area of a circle is defined as:

$A=\pi r^2$

For your data, this is:

$45 = \pi r^2$

To solve for , first divide both sides by $\pi$ . Then take the square root of both sides. Thus you get:

r =3.78469878303024...

The diameter of the circle is just double that:

d=7.56939756606048

Rounding to the nearest hundredth, you get 7.57 .

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Example Question #1 : Diameter And Chords

Let represent the area of a circle and represent its circumference. Which of the following equations expresses in terms of ?

Possible Answers:

C^2

Correct answer:

Explanation:

The formula for the area of a circle is $A=\pi r^2$ , and the formula for circumference is $C=2\pi r$ . If we solve for C in terms of r, we get
$r=C/2\pi$ .

We can then substitute this value of r into the formula for the area:

$A=\pi r^2$

$=\pi (C/2\pi )^2$

$=C^2\pi /4\pi ^2$

$=C^2/4\pi$

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Example Question #531 : Plane Geometry

A circle has a radius of . What is the ratio of its circumference to its area?

Possible Answers:

$4\pi$

$2\pi$

$\pi$

Correct answer:

Explanation:

1. Find the circumference:

$C=\pi d=2\pi r$

$C=2\cdot \pi \cdot 2=4\pi$

2. Find the area:

$A=\pi r^{2}$

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

3. Divide the circumference by the area:

$\frac{C}{A}=\frac{4\pi }{4\pi }=1$

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Example Question #532 : Plane Geometry

How far must a racehorse run in a lap race where the course is a circle with an area of $36\pi$ ?

Possible Answers:

$24\pi$

$36\pi$

$48\pi$

$6\pi$

$12\pi$

Correct answer:

$48\pi$

Explanation:

To find the answer, we need to first find the circumference of the circle then mulitply that by 4 because the racehorse is running 4 laps.

We know that the area of the circle is $36\pi$ . We can then find the radius.

$A=\pi r^2$

$36\pi=\pi r^2$

r^2=36

r=6

Because d=2r , our diameter is 12.

We can now find the circumference.

$C=d\pi$

$C=12 \pi$

Multiply the circumference by 4 (our racehorse is running 4 laps) to find the answer.

$12 \pi \cdot 4=48\pi$

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

If the equation of a circle is (x – 7)²+ (y + 1)²= 81, what is the area of the circle?

Possible Answers:

81π

6561π

49π

2π

18π

Correct answer:

81π

Explanation:

The equation is already in a circle equation, and the right side of the equation stands for r^2 →r²= 81 and r = 9

The area of a circle is πr², so the area of this circle is 81π.

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

Assume π = 3.14

A man would like to put a circular whirlpool in his backyard. He would like the whirlpool to be six feet wide. His backyard is 8 feet long by 7 feet wide. By state regulation, in order to put a whirlpool in a backyard space, the space must be 1.5 times bigger than the pool. Can the man legally install the whirlpool?

Possible Answers:

No, because the area of the backyard is 30 square feet and therefore the whirlpool is too big to meet the legal requirement.

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

Yes, because the area of the whirlpool is 18.84 square feet and 1.5 times its area would be less than the area of the backyard.

No, because the area of the backyard is smaller than the area of the whirlpool.

No, because the area of the whirlpool is 42.39 square feet and 1.5 times its area would be greater than the area of the backyard.

Correct answer:

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

Explanation:

If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.

If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.

If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.

And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.

The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet.

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #4 : How To Find The Length Of The Diameter

Example Question #1 : Diameter

Example Question #2 : Diameter

Example Question #6 : How To Find The Length Of The Diameter

Example Question #2 : How To Find The Length Of The Diameter

Example Question #1 : Diameter And Chords

Example Question #531 : Plane Geometry

Example Question #532 : Plane Geometry

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

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

All ACT Math Resources

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