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ACT Math : Plane Geometry

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Example Questions

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?

Possible Answers:

2√2 in²

8π in²

32π in²

16π in²

4√2 in²

Correct answer:

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.

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Example Question #211 : Problem Solving

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

1936π dollars

1936 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.

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Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

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

625 dollars

500 dollars

2500 - 625π dollars

500 - 125π dollars

Correct answer:

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.

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Example Question #11 : Radius

Possible Answers:

50π

20π

10π

25π

100π

Correct answer:

50π

Explanation:

Circle_graph_area2

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Example Question #551 : Plane Geometry

A circle with diameter of length $\dpi{100} d$ 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:

$d^{2}-4$

$\frac{(4-\pi)d^2}{4}$

$\frac{(\pi - 1)d^2}{4}$

$\pi d^{2}+2$

$(\pi - 1)d^2$

Correct answer:

$\frac{(4-\pi)d^2}{4}$

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 $\pi r^2$ . 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 $\dpi{100} r$ :

$\dpi{100} d=2r$

Divide both sides by 2.

$r = \frac{d}{2}$

We will now substitute this into the formula for the area of the circle.

area of circle = $\pi r^2 = \pi(\frac{d}{2})^2=\pi \frac{d}{2}\cdot \frac{d}{2}= \frac{\pi d^2}{4}$

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 $d^{2}$ .

area of square = $d^{2}$

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

difference in areas = $d^2 - \frac{\pi d^2}{4}$

We will rewrite $d^{2}$ so that its denominator is 4.

difference in areas = $\frac{4d^2}{4}-\frac{\pi d^2}{4}=\frac{(4-\pi)d^2}{4}$

The answer is $\frac{(4-\pi)d^2}{4}$ .

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Example Question #71 : Circles

An original circle has an area of $16\pi$ . If the radius is increased by a factor of 3, what is the ratio of the new area to the old area?

Possible Answers:

$3:1$

$7:1$

$9:1$

$10:1$

$8:1$

Correct answer:

$9:1$

Explanation:

The formula for the area of a circle is $\pi r^{2}$ . If we increase $r$ by a factor of 3, we will increase the area by a factor of 9.

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Example Question #14 : Radius

A square has an area of $36\ in^{2}$ . 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:

$27\pi \ in^{2}$

$36\pi \ in^{2}$

$12\pi \ in^{2}$

$9\pi \ in^{2}$

$18\pi \ in^{2}$

Correct answer:

$9\pi \ in^{2}$

Explanation:

The area of a square is given by $A = s^{2}$ 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 $A = \pi r^{2}$ or $9\pi \ in^{2}$ .

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Example Question #71 : 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?

Possible Answers:

$191\pi\hspace{1 mm}inches^2$

$216\hspace{1 mm}inches^2$

$216-25\pi\hspace{1 mm}inches^2$

$25\pi\hspace{1 mm}inches^2$

$216\pi\hspace{1 mm}inches^2$

Correct answer:

$216-25\pi\hspace{1 mm}inches^2$

Explanation:

First we will calculate the total area of the placemat:

$A=l\times w= 18\times 12= 216\hspace{1 mm}inches^2$

Next we will calculate the area of the circular place

$A=\pi r^2$

And

$d=2r=10$

$r=5\hspace{1 mm}inches$

$A=\pi r^2=\pi (5^2)=25\pi\hspace{1 mm}inches^2$

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

$216-25\pi\hspace{1 mm}inches^2$

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Example Question #552 : Plane Geometry

The picture above contains both a circle with diameter 4, and a rectangle with length 8 and width 5. Find the area of the shaded region. Round your answer to the nearest integer

Possible Answers:

Correct answer:

Explanation:

First, recall that the diameter of a circle is twice the value of the radius. Therefore a circle with diameter 4 has a radius of 2. Next recall that the area of a circle with radius r=2 is:

$\pi r^2 = \pi(2)^2 = 4 \pi$

The area of the rectangle is the length times the width:

$L \times W = 8 \times 5 = 40$

The area of the shaded region is the difference between the 2 areas:

$40 - 4 \pi \approx 40-4(3.14) = 27.44$

The nearest integer is 27.

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Example Question #21 : Radius

Allen was running around the park when he lost his keys. He was running around aimlessly for the past 30 minutes. When he checked 10 minutes ago, he still had his keys. Allen guesses that he has been running at about 3m/s.

If Allen can check 1 square kilometer per hour, what is the longest it will take him to find his keys?

Possible Answers:

$\sim 81\ hours$

$\sim 10\ hours$

$\sim 100\ hours$

$\sim 1,000\ hours$

$\sim 3\ hours$

Correct answer:

$\sim 10\ hours$

Explanation:

Allen has been running for 10 minutes since he lost his keys at 3m/s. This gives us a maximum distance of 3*600=1800m from his current location. If we move 1800m in all directions, this gives us a circle with radius of 1800m. The area of this circle is

$1800^2\pi \approx 10,178,760m^2$

Our answer, however, is asked for in kilometers. 1800m=1.8km, so our actual area will be $\frac{10,178,760}{1,000,000} = 10.2$ square kilometers. Since he can search 1 per hour, it will take him at most 10.2 hours to find his keys.

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

Example Question #211 : Problem Solving

Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

Example Question #11 : Radius

Example Question #551 : Plane Geometry

Example Question #71 : Circles

Example Question #14 : Radius

Example Question #71 : Circles

Example Question #552 : Plane Geometry

Example Question #21 : Radius

All ACT Math Resources

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