### All ACT Math Resources

## Example Questions

### Example Question #51 : Circles

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

**Possible Answers:**

50π

25π

10π

40π

100π

**Correct answer:**

25π

Area of a circle = A = πr^{2}

R = 1/2d = ½(10) = 5

A = 5^{2}π = 25π

### Example Question #11 : Radius

A square has an area of 1089 in^{2}. If a circle is inscribed within the square, what is its area?

**Possible Answers:**

1089π in^{2}

33π in^{2}

33 in^{2}

16.5 in^{2}

272.25π in^{2}

**Correct answer:**

272.25π in^{2}

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 = s^{2} or 1089 = s^{2}. 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 πr^{2} or π16.5^{2} = 272.25π.

### Example Question #22 : Radius

A square has an area of 32 in^{2}. If a circle is inscribed within the square, what is its area?

**Possible Answers:**

4√2 in^{2}

2√2 in^{2}

16π in^{2}

32π in^{2}

8π in^{2}

**Correct answer:**

8π in^{2}

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 = s^{2} or 32 = s^{2}. Taking the square root of both sides, we get: s = √32 = √(2^{5}) = 4√2.

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

### Example Question #11 : Radius

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:**

7744 – 1936π dollars

1936π dollars

5808 dollars

1936 – 484π dollars

1936 dollars

**Correct answer:**

1936 – 484π dollars

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 in^{2}.

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr^{2} or π * 44^{2} = 1936π in^{2}. 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 #12 : 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.?

**Possible Answers:**

625 - 25π dollars

500 dollars

625 dollars

2500 - 625π dollars

500 - 125π dollars

**Correct answer:**

500 - 125π dollars

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 in^{2}.

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr^{2} or π * 25^{2} = 625π in^{2}. 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 #12 : Radius

**Possible Answers:**

50*π*

20*π*

100*π*

25*π*

10*π*

**Correct answer:**

50*π*

### Example Question #311 : Plane Geometry

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?

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #312 : Plane Geometry

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?

**Possible Answers:**

**Correct answer:**

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 #21 : Radius

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?

**Possible Answers:**

**Correct answer:**

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 #21 : Basic Geometry

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?

**Possible Answers:**

**Correct answer:**

First we will calculate the total area of the placemat:

Next we will calculate the area of the circular place

And

So

We will subtract the area of the plate from the total area

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