High School Math : Circles

Example Questions

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

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?

25π in2

55π in2

72π in2

28π in2

7π in2

55π in2

Explanation:

The area of a circle is πr2

The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 32 = 9π in2

The radius of the second circle = 8” so the area = π * 82 = 64π in2

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

Area = 64π in2  - 9π in2 = 55π in2

Example Question #5 : How To Find The Area Of A Circle

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

12

8

4

6

7

8

Explanation:

The area of a circle is A=πr2. 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 #11 : How To Find The Area Of A Circle

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?

3

9

15

12

6

9

Explanation:

We know that the equation for the area of a circle is π r2. 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 62)/(π 22) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.

Example Question #12 : How To Find The Area Of A Circle

1. A circle is inscribed inside a 10 by 10 square. What is the area of the circle?

25π

100π

50π

40π

10π

25π

Explanation:

Area of a circle = A = πr2

R = 1/2d = ½(10) = 5

A = 52π = 25π

Example Question #13 : How To Find The Area Of A Circle

A square has an area of 1089 in2. If a circle is inscribed within the square, what is its area?

33π in2

16.5 in2

272.25π in2

1089π in2

33 in2

272.25π in2

Explanation:

The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:

A = s2 or 1089 = s2. Taking the square root of both sides, we get: s = 33.

Now, based on this, we know that 2r = 33 or r = 16.5. The area of the circle is πr2 or π16.52 = 272.25π.

Example Question #14 : How To Find The Area Of A Circle

A square has an area of 32 in2. If a circle is inscribed within the square, what is its area?

8π in2

32π in2

4√2 in2

16π in2

2√2 in2

8π in2

Explanation:

The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:

A = s2 or 32 = s2. Taking the square root of both sides, we get: s = √32 = √(25) = 4√2.

Now, based on this, we know that 2r = 4√2 or r = 2√2. The area of the circle is πr2 or π(2√2)2 = 4 * 2π = 8π.

A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.25 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 44 in.? Possible Answers: 5808 dollars 1936 dollars 1936π dollars 7744 – 1936π dollars 1936 – 484π dollars Correct answer: 1936 – 484π dollars Explanation: The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 88 inches for our problem. Its total area would be 88 * 88 or 7744 in2. Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 442 = 1936π in2. Therefore, the area remaining would be 7744 – 1936π. The cost of the waste would be 0.25 * (7744 – 1936π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 4 from our subtraction. This would give us: 0.25 * 4 * (1936 – 484π). Since 0.25 is equal to 1/4, 0.25 * 4 = 1. Therefore, our final answer is: 1936 – 484π dollars. Example Question #16 : How To Find The Area Of A Circle A manufacturer makes wooden circles out of square blocks of wood. If the wood costs$0.20 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 25 in.?

500 dollars

625 dollars

625 - 25π dollars

2500 - 625π dollars

500 - 125π dollars

500 - 125π dollars

Explanation:

The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 50 inches for our problem. Its total area would be 50 * 50 or 2500 in2.

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 252 = 625π in2. Therefore, the area remaining would be 2500 - 625π. The cost of the waste would be 0.2 * (2500 – 625π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 5 from our subtraction. This would give us: 0.2 * 5 * (500 – 125π). Since 0.2 is equal to 1/5, 0.2 * 625 = 125. Therefore, our final answer is: 500 – 125π dollars.

Example Question #52 : Circles 50π

25π

100π

10π

20π

50π

Explanation: Example Question #18 : How To Find The Area Of A Circle

A circle with diameter of length is inscribed in a square. Which of the following is equivalent to the area inside of the square, but outside of the circle?      Explanation:

In order to find the area that is inside the square but outside the circle, we will need to subtract the area of the circle from the area of the square. The area of a circle is equal to . However, since we are given the length of the diameter, we will need to solve for the radius in terms of the diameter. Because the diameter of a circle is twice the length of its radius, we can write the following equation and solve for : Divide both sides by 2. We will now substitute this into the formula for the area of the circle.

area of circle = We next will need to find the area of the square. Because the circle is inscribed in the square, the diameter of the circle is equal to the length of the circle's side. In other words, the square has side lengths equal to d. The area of any square is equal to the square of its side length. Therefore, the area of the square is .

area of square = Lastly, we will subtract the area of the circle from the area of the square.

difference in areas = We will rewrite so that its denominator is 4.

difference in areas = The answer is . 