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# Area of a Circle

## Pi times the radius squared.

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Area of a Circle

What if you wanted to figure out the area of a circle with a radius of 5 inches? After completing this Concept, you'll be able to answer questions like this.

### Guidance

Recall that \begin{align*}\pi\end{align*} is the ratio between the circumference of a circle and its diameter. We are going to use the formula for circumference to derive the formula for area.

First, take a circle and divide it up into several wedges, or sectors. Then, unfold the wedges so they are all on one line, with the points at the top.

Notice that the height of the wedges is \begin{align*}r\end{align*}, the radius, and the length is the circumference of the circle. Now, we need to take half of these wedges and flip them upside-down and place them in the other half so they all fit together.

Now our circle looks like a parallelogram. The area of this parallelogram is \begin{align*}A=bh=\pi r \cdot r=\pi r^2\end{align*}.

To see an animation of this derivation, see http://www.rkm.com.au/ANIMATIONS/animation-Circle-Area-Derivation.html, by Russell Knightley.

The formula for the area of a circle is \begin{align*}A=\pi r^2\end{align*} where \begin{align*}r\end{align*} is the radius of the circle.

#### Example A

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

If the diameter is 12 cm, then the radius is 6 cm. The area is \begin{align*}A=\pi (6^2)=36 \pi \ cm^2\end{align*}.

#### Example B

If the area of a circle is \begin{align*}20 \pi\end{align*}, what is the radius?

Work backwards on this problem. Plug in the area and solve for the radius.

\begin{align*}20 \pi &= \pi r^2\\ 20 &= r^2\\ r &= \sqrt{20}=2 \sqrt{5}\end{align*}

Just like the circumference, we will leave our answers in terms of \begin{align*}\pi\end{align*}, unless otherwise specified. In Example 2, the radius could be \begin{align*}\pm 2 \sqrt{5}\end{align*}, however the radius is always positive, so we do not need the negative answer.

#### Example C

A circle is inscribed in a square. Each side of the square is 10 cm long. What is the area of the circle?

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

\begin{align*}A=\pi 5^2=25 \pi \ cm\end{align*}

Watch this video for help with the Examples above.

#### Concept Problem Revisited

A circle with a radius of 5 inches has area \begin{align*}\pi5^2 = 25\pi \ in^2 \end{align*}.

### Vocabulary

A circle is the set of all points that are the same distance away from a specific point, called the center. A radius is the distance from the center to the outer rim of the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. Area is the amount of space inside a figure and is measured in square units. \begin{align*}\pi\end{align*}, or pi is the ratio of the circumference of a circle to its diameter.

### Guided Practice

1. Find the area of the shaded region from Example C.

2. Find the diameter of a circle with area \begin{align*}36 \pi\end{align*}.

3. Find the area of a circle with diameter 20 inches.

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

\begin{align*}A=10^2-25 \pi =100-25 \pi \approx 21.46 \ cm^2\end{align*}

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

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

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

3. If the diameter is 20 inches that means that the radius is 10 inches. Now we can use the formula for the area of a circle. \begin{align*}A=\pi (10)^2=100\pi \ in^2\end{align*} .

### Practice

Fill in the following table. Leave all answers in terms of \begin{align*}\pi\end{align*}.

1. 2
2. \begin{align*}16 \pi\end{align*}
3. \begin{align*}10\end{align*}
4. \begin{align*}24\end{align*}
5. 9
6. \begin{align*}90 \pi\end{align*}
7. \begin{align*}35\end{align*}
8. \begin{align*}\frac{7}{\pi}\end{align*}
9. 60
10. \begin{align*}36 \pi\end{align*}

1. Carlos has 400 ft of fencing to completely enclose an area on his farm for an animal pen. He could make the area a square or a circle. If he uses the entire 400 ft of fencing, how much area is contained in the square and the circle? Which shape will yield the greatest area?

### Vocabulary Language: English Spanish

chord

A line segment whose endpoints are on 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.