### All ACT Math Resources

## Example Questions

### Example Question #1 : Radius

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

**Possible Answers:**

6561π

18π

81π

2π

49π

**Correct answer:**

81π

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

The area of a circle is πr^{2}, so the area of this circle is 81π.

### Example Question #1 : Plane Geometry

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:**

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.

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

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 30 square feet and therefore the whirlpool is too big to meet the legal requirement.

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.

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.

### Example Question #2 : Plane Geometry

There are two identical circles on a plane that overlap. The radius of both circles is 1. The region in which they overlap has an area of *π*.

What is the total area of the 2 overlapping circles?

**Possible Answers:**

2*π*

*π*

2

0

1

**Correct answer:**

*π*

The total area of both circles is *π* + *π* = 2*π*

Since the region overlaps, we cannot count it twice, so we must subtract it.

we get 2*π* – *π* = *π*

### Example Question #1 : Plane Geometry

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

**Possible Answers:**

8π - 16

4π-4

8π-4

8π-8

2π-4

**Correct answer:**

8π - 16

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)^{2}, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

### Example Question #1 : Radius

If a circle has a circumference of 16π, what would its area be if its radius were halved?

**Possible Answers:**

4π

16π

8π

64π

**Correct answer:**

16π

The circumference of a circle = πd where d = diameter. Therefore, this circle’s diameter must equal 16. Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8. Halving the radius would give us a new radius of 4. To find the area of this new circle, use the formula A=πr² where r = radius. Plug in 4 for r. Area will equal 16π.

### Example Question #11 : How To Find The Area Of A Circle

A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.

**Possible Answers:**

361.5

351.5

346.5

341.5

356.5

**Correct answer:**

361.5

The area of the circle is (30/2)^{2}*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5

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

The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?

**Possible Answers:**

64π

49π

160π

54π

50π

**Correct answer:**

64π

The original radius would be 4, making the new radius 8 and by the area of a circle (A=π(r)^{2}) the new area would be 64π.

### Example Question #1 : Radius

A circle with a diameter of 6” sits inside a circle with a radius of 8”. What is the area of the interstitial space between the two circles?

**Possible Answers:**

25π in2

28π in2

7π in2

55π in2

72π in2

**Correct answer:**

55π in2

The area of a circle is πr^{2}.

The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 3^{2} = 9π in^{2}

The radius of the second circle = 8” so the area = π * 8^{2} = 64π in^{2}

The area of the interstitial space = area of the first circle – area of the second circle.

Area = 64π in^{2 }- 9π in^{2} = 55π in^{2}

### Example Question #11 : Circles

If the radius of a circle is tripled, and the new area is 144π what was the diameter of the original circle?

**Possible Answers:**

12

4

7

6

8

**Correct answer:**

8

The area of a circle is A=πr^{2}. Since the radius was tripled 144π =π(3r)^{2}. Divide by π and then take the square root of both sides of the equal sign to get 12=3r, and then r=4. The diameter (d) is equal to twice the radius so d= 2(4) = 8.

### Example Question #542 : Act Math

If the radius of Circle A is three times the radius of Circle B, what is the ratio of the area of Circle A to the area of Circle B?

**Possible Answers:**

3

12

6

15

9

**Correct answer:**

9

We know that the equation for the area of a circle is π r^{2}. To solve this problem, we pick radii for Circles A and B, making sure that Circle A’s radius is three times Circle B’s radius, as the problem specifies. Then we will divide the resulting areas of the two circles. For example, if we say that Circle A has radius 6 and Circle B has radius 2, then the ratio of the area of Circle A to B is: (π 6^{2})/(π 2^{2}) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.