Example Questions

Example Question #541 : Act Math

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

40π

50π

25π

10π

100π

25π

Explanation:

Area of a circle = A = πr2

R = 1/2d = ½(10) = 5

A = 52π = 25π

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

1089π in2

33π in2

272.25π in2

33 in2

16.5 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 #11 : 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?

32π in2

2√2 in2

4√2 in2

16π in2

8π 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 #661 : Grade 7 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.?

625 dollars

500 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.

10π

25π

100π

50π

20π

50π

Explanation:

Example Question #86 : Circles

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 =

Example Question #87 : Circles

An original circle has an area of . If the radius is increased by a factor of 3, what is the ratio of the new area to the old area?

Explanation:

The formula for the area of a circle is . If we increase  by a factor of 3, we will increase the area by a factor of 9.

Example Question #81 : Circles

A square has an area of .  If the side of the the square is the same as the diameter of a circle, what is the area of the circle?

Explanation:

The area of a square is given by so we know that the side of the square is 6 in.  If a circle has a diameter of 6 in, then the radius is 3 in.  So the area of the circle is   or .

Example Question #89 : Circles

Mary has a decorative plate with a diameter of ten inches. She places the plate on a rectangular placemat with a length of 18 inches and a width of 12 inches. How much of the placemat is visible?