# 6.6: Circle Area

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## Area of a Circle

To find the area of a circle, all you need to know is its radius. If r is the radius of a circle, then its area is $$A=\pi r^2$$. Figure $$\PageIndex{1}$$

We will leave our answers in terms of $$\pi$$, unless otherwise specified.

What if you were given the radius or diameter of a circle? How could you find the amount of space the circle takes up?

Example $$\PageIndex{1}$$

Find the area of a circle with a diameter of 12 cm.

Solution

If $$d=12\text{ cm }$$, then $$r=6\text{ cm }$$. The area is $$A=\pi (6^2)=36\pi \text{ cm }^2$$.

Example $$\PageIndex{2}$$

If the area of a circle is $$20\pi \text{ units }$$, what is the radius?

Solution

Plug in the area and solve for the radius.

\begin{aligned} 20\pi &=\pi r^2 \\ 20&=r^2 \\ r&=\sqrt{20}=2\sqrt{5}\text{ units } \end{aligned}

Example $$\PageIndex{3}$$

A circle is inscribed in a square. Each side of the square is 10 cm long. What is the area of the circle? Figure $$\PageIndex{2}$$

Solution

The diameter of the circle is the same as the length of a side of the square. Therefore, the radius is 5 cm.

$$A=\pi 5^2=25\pi \text{ cm }^2$$

Example $$\PageIndex{4}$$

Find the area of the shaded region from Example 3.

Solution

The area of the shaded region would be the area of the square minus the area of the circle.

$$A=102−25\pi =100−25\pi \approx 21.46\text{ cm }^2$$

Example $$\PageIndex{5}$$

Find the diameter of a circle with area $$36\pi$$.

Solution

First, use the formula for the area of a circle to solve for the radius of the circle.

\begin{aligned}A&=\pi r^2 \\ 36\pi &=\pi r^2 \\ 36&=r^2 \\ r&=6\end{aligned}

If the radius is 6 units, then the diameter is 12 units.

## Review

Fill in the following table. Leave all answers in terms of $$\pi$$.

1. 2
2. $$16\pi$$
3. $$10\pi$$
4. $$24\pi$$
5. 9
6. $$90\pi$$
7. $$35\pi$$
8. $$7\pi$$
9. 60
10. 36

1. Figure $$\PageIndex{3}$$
2. Figure $$\PageIndex{4}$$
3. Figure $$\PageIndex{5}$$

## Vocabulary

Term Definition
chord A line segment whose endpoints are on a circle.
circle The set of all points that are the same distance away from a specific point, called the center.
circumference The distance around a circle.
diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
pi (or $$\pi$$ ) The ratio of the circumference of a circle to its diameter.
radius The distance from the center to the outer rim of a circle.

Interactive Element

Video: Determine the Area of a Circle

Activities: Area of a Circle Discussion Questions

Study Aids: Circumference and Arc Length Study Guide

Practice: Circle Area

Real World: Ringside Seats

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