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High School Math : Circles

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Example Questions

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 #81 : Circles

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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Example Question #81 : Plane Geometry

To the nearest tenth, give the diameter of a circle with area 100 square inches.

Possible Answers:

$11.3 \textrm{ in}$

$20.0 \textrm{ in}$

$5.7 \textrm{ in}$

$22.6 \textrm{ in}$

$31.4 \textrm{ in}$

Correct answer:

$11.3 \textrm{ in}$

Explanation:

The relationship between the radius and the area of a circle can be given as

$A = \pi r^{2}$ .

We can substitute A = 100 and solve for :

$100= \pi r^{2}$

$r^{2} = \frac{100}{\pi}$

$r=\sqrt{ \frac{100}{\pi}} \approx 5.64$

Double this to get the diameter: $d \approx 5.64 \cdot2 =11.28$ , which we round to 11.3.

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Example Question #35 : Radius

To the nearest tenth, give the area of a circle with diameter $7 \frac{1}{2}$ inches.

Possible Answers:

$88.4 \textrm{ in}^{2}$

$56.3 \textrm{ in}^{2}$

$44.2 \textrm{ in}^{2}$

$22.1 \textrm{ in}^{2}$

$176.7 \textrm{ in}^{2}$

Correct answer:

$44.2 \textrm{ in}^{2}$

Explanation:

The radius of a circle with diameter $7 \frac{1}{2}$ inches is half that, or $3 \frac{3}{4} = 3.75$ inches. The area of the circle is

$A = \pi r^{2} = \pi \cdot 3.75^{2} \approx 44.2 \textrm{ in}^{2}$

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Example Question #61 : Circles

To the nearest tenth, give the area of a circle with diameter 17 inches.

Possible Answers:

$454.0\textrm{ in}^{2}$

$289.0 \textrm{ in}^{2}$

$907.9\textrm{ in}^{2}$

$227.0 \textrm{ in}^{2}$

$578.0\textrm{ in}^{2}$

Correct answer:

$227.0 \textrm{ in}^{2}$

Explanation:

The radius of a circle with diameter 17 inches is half that, or 8.5 inches. The area of the circle is

$A = \pi r^{2} = \pi \cdot 8.5^{2} \approx 227.0$

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Example Question #92 : Plane Geometry

A circle has a radius of . A second circle has a radius of . What is the ratio of the larger circle's area to the smaller circle's area?

Possible Answers:

$\pi : 1$

10:1

8:1

3:1

5:1

Correct answer:

5:1

Explanation:

The area of a circle is given by the equation $A = \pi r^2$ , where is the area and is the radius of the circle. Use this formula to determine the areas of the two circles:

$A = \pi(15)^2$ and $A = \pi(5)^2$

$\rightarrow A = 125\pi$ units squared and $A = 25\pi$ units squared.

The ratio of the larger circle to the smaller circle is $125\pi:25\pi$ . Divide each side of the ratio by $25\pi$ to express it in its simplest form, 5:1 .

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Example Question #62 : Circles

What is the area of a circle with a radius of $\sqrt{3}$ ?

Possible Answers:

$9\Pi$

$3\Pi$

$\sqrt{3}\Pi$

$6\Pi$

$\sqrt{3\Pi }$

Correct answer:

$3\Pi$

Explanation:

The equation for the area of a circle is $\Pi r^2$ .

By substituing the given radius of $\sqrt{3}$ into the equation, we get $3\Pi$ .

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High School Math : Circles

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #71 : Circles

Example Question #14 : Radius

Example Question #71 : Circles

Example Question #81 : Circles

Example Question #21 : Radius

Example Question #81 : Plane Geometry

Example Question #35 : Radius

Example Question #61 : Circles

Example Question #92 : Plane Geometry

Example Question #62 : Circles

All High School Math Resources

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