# Quarter Circles: Calculating Area, Perimeter & Radius

## A Quarter Circle

Meet the **quarter circle**.

It is one fourth of a circle. If you take a whole circle and slice it into four pieces, then one of those slices makes a quarter circle.

In this lesson, we will look at finding the area, the perimeter, and the radius of a quarter circle. Knowing how to find these key pieces of information for quarter circles will serve you well as you advance in your math classes and as you take standardized math tests. Not only will you find these types of problems on math tests, but you may also find these types of problems working as an architect or as an engineer.

So let's get started.

## Calculating Area

First, let's talk about area. Normally, with these types of problems, you will be given the radius of the circle. Look carefully at your quarter circle and you will see that the two straight sides are both a radius of the circle. So, you only need the measurement of one of these straight sides. The other side will be the same.

Let's work through a problem together. Let's say the problem tells you that one of the straight sides of the quarter circle measures 3 inches, and it wants you to find the area of the quarter circle. How would you go about finding the answer?

Well, you think about what the quarter circle is. It is one fourth of a whole circle. So, you can find the area of the whole circle and then divide by 4 to find the area of your quarter circle. You know that the area of a whole circle is found by using the formula *A = pi * r^2*, where you square your radius and multiply it by pi. You can do the same for this problem and then divide this number by 4 to find your answer.

Using the formula and plugging in 3 for *r*, you get *A* = pi * 3^2 = 9 * pi = 28.26 inches squared. So the area of the whole circle is 28.26 inches squared. Now, to find the area of the quarter circle, you divide the 28.26 by 4 to get 28.26 / 4 = 7.065 inches squared. So your answer is 7.065 inches squared.

Remember that your area units are always squared. That is why your answer includes the inches squared at the end.

## Calculating Perimeter

Now, let's calculate the perimeter. We will work off the same problem where the radius is 3 inches. You might think that all you have to do is to divide the perimeter or circumference of the whole circle by 4 to find your answer, kind of like what you did for the area. You are close, but you are missing part of your answer.

Look carefully at your quarter circle: what do you see in addition to a quarter of the circumference of the whole circle? Remember the circumference of the whole circles gives you that outer ring. What else do you have in your quarter circle that you don't have in your whole circle? That's right, you have the two straight sides.

So, to find your circumference, you can find a quarter of the circumference of the whole circle first. This will give you the length of the curve. Then you can add the two straight sides to find your answer. Remember, your two straight sides are both the radii of your circle, so their lengths will be the same as the length of the radius that is given.

For our problem, you remember that the formula for finding the perimeter or circumference of a whole circle is *C = 2 * pi * r*. You multiply the radius by 2 and pi. For our answer, we will again leave it in simplified fraction form and with the pi. So, your radius measures 3, so the circumference of the whole circle is *C* = 2 * pi * 3 = 6 * pi = 18.84 inches. Dividing this by 4 gives you 18.84 / 4 inches = 4.71 inches. This is only the curved part, so now you add the straight sides to this measurement.

You have two straight sides that both measure 3, so you add 4.71 + 3 + 3 = 10.71 inches. So, your perimeter of your quarter circle is 10.71 inches.

## Calculating Radius

Let's talk about finding the radius now. The first two problems gave you the radius. Now, what if the problem gave you just the area of the quarter circle or just the length of the curved part of the quarter circle, and then asked you to find the radius? How do you solve these types of problems? You would work backwards.

If the problem gave you the area, you first multiply it by 4 to give you the area of the whole circle. And then you can plug this number in for *A* in the formula *A = pi * r^2* and then solve for *r*. Let's look at an example.

Let's say the given area is 7.065 inches squared for the quarter circle. To find the radius, you first multiply the 7.065 by 4. You get 7.065 * 4 = 28.26. Now you plug this into the formula *A = pi * r^2* for *A*. You have 28.26 = pi * r^2. Solving for *r*, you first divide the 28.26 by pi and then you take the square root. So, dividing by pi or 3.14, you get 28.26 / 3.14 = 9. The square root of 9 is 3. So the radius is 3 inches here.

Now, if the problem only gave you the length of the curved part, you again would multiply it by 4 to find the circumference of the whole circle. You then plug in the circumference of the whole circle into the formula *C = 2 * pi * r* for *C* and then solve for *r*.

So, say the curved part measures 4.71 inches. To find the radius of the quarter circle, you first multiply the 4.71 by 4. You get 4.71 * 4 = 18.84 inches. Now, you plug this in for *C* into the formula *C = 2 * pi * r*. You get 18.84 = 2 * pi * r. Solving for *r*, you divide the 18.84 by 2 * pi or 2 * 3.14 = 6.28. Doing that, you get 18.84 / 6.28 = 3 inches. So your answer is 3 inches.

## Lesson Summary

Let's review what you've learned. A **quarter circle** is one fourth of a circle. To find the area of a quarter circle, find the area of the whole circle by using the formula *A = pi * r^2* and then divide by 4. To find the perimeter of the quarter circle, find the circumference of the whole circle, divide by 4, and then add the radius twice.

To find the radius when you are given the area of the quarter circle, multiply the given area by 4, then plug this number in for *A* into the formula *A = pi * r^2* and then solve for *r*. To find the radius when you are given the length of the curved part of the quarter circle, multiply this length by 4 and then plug in this number into the formula *C = 2 * pi * r* for *C*. Then solve for *r*.

## Learning Outcomes

When you are done, you're ready to:

- Identify a quarter circle
- Calculate the area, perimeter, radius, and circumference of a quarter circle

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## Practice Questions

1. A quarter circle has area = 26π square centimeters (cm^2). Find the radius of this quarter circle.

2. Find the whole perimeter of a quarter circle whose radius is 5 inches.

## Answers

1. The area A of a quarter circle of radius r is one-fourth of the area of the whole circle of radius r. Hence the area of the quarter circle will be given by the following formula

The question gives us A = 26π and asks us to find r. Using the formula above, we get

Multiplying both sides of the above equation by 4 and then dividing both sides by π yields the radius of the quarter circle as

2. The perimeter P of a whole quarter circle equals the perimeter of the curved part of the circle, which is one fourth the perimeter of the whole circle, added to 2 times the radius r of the circle.

Hence,

Since r = 5 inches from the question, we substitute in the above formula to get the whole perimeter of the quarter circle as

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