How does the area of a circle relate to the area of a rectangle?

#### Watch This

http://www.youtube.com/watch?v=SIKkWLqt2mQ James Sousa: Determine the Area of a Circle

http://www.youtube.com/watch?v=sHtsnC2Mgnk James Sousa: Determine the Circumference of a Circle

#### Guidance

A circle is the set of all points equidistant from a center point. From the center point to the circle is called the **radius** \begin{align*}(r)\end{align*}**diameter** \begin{align*}(d)\end{align*}

The perimeter of a circle is called its **circumference**. The ratio between the circumference and diameter of any circle is \begin{align*}\pi\end{align*}

\begin{align*}C=\pi d\end{align*}

The area of a circle is also related to \begin{align*}\pi\end{align*}

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

**Example A**

Find the area and circumference of a circle with radius 5 inches.

**Solution:**

\begin{align*}Area =\pi r^2=\pi(5^2)=25 \pi \ in^2\end{align*}

\begin{align*}Circumference =\pi d=\pi(2.5)=10 \pi \ in\end{align*}

Leaving your answer with a \begin{align*}\pi\end{align*}

**Example B**

Find the area and circumference of a circle with diameter 16 cm.

**Solution:** If the diameter is 16 cm, then the radius is 8 cm.

\begin{align*}Area =\pi r^2=\pi(8^2)=64 \pi \ cm^2\end{align*}

\begin{align*}Circumference =\pi d=16 \pi \ cm\end{align*}

**Example C**

The shape below is a portion of a circle called a sector. This sector is \begin{align*}\frac{1}{4}\end{align*}

**Solution:** Since the sector is \begin{align*}\frac{1}{4}\end{align*}

\begin{align*}Area_{Sector}=\frac{\pi(4^2)}{4}=4 \pi \ in^2\end{align*}

The perimeter of the sector is the sum of the lengths of the two radiuses and the arc. Each radius is 4 *in*. The arc is \begin{align*}\frac{1}{4}\end{align*}

\begin{align*}Perimeter_{Sector}=4+4+2 \pi=8+2 \pi \ in\end{align*}

**Concept Problem Revisited**

You can understand the formula for the area of a circle by dissecting a circle into wedges and rearranging them to form a shape that is close to a parallelogram. The parallelogram can then by formed into a shape close to a rectangle.

The lengths of the sides of the “parallelogram” are \begin{align*}r\end{align*}

#### Vocabulary

A ** circle** is the set of all points equidistant from a center point.

** Equidistant** means the same distance away from.

** Circumference** is the perimeter of a circle.

\begin{align*}\pi\end{align*}

The ** radius** of a circle is the distance from the center point to the circle.

The ** diameter** of a circle is the distance from one side of the circle to the other while passing through the center point.

#### Guided Practice

1. Explain why the formula \begin{align*}C=2 \pi r\end{align*}

2. Find the area of a circle with a circumference of \begin{align*}12 \pi \ cm\end{align*}

3. Find the area and circumference of the sector below. The sector is \begin{align*}\frac{1}{3}\end{align*}

**Answers:**

1. The relationship between the radius and the diameter is \begin{align*}d=2r\end{align*}

2. If the circumference is \begin{align*}12 \pi \ cm\end{align*}

3. \begin{align*}Area=\frac{\pi r^2}{3}=\frac{\pi(9^2)}{3}=27 \pi \ cm^2\end{align*}.

\begin{align*}Perimeter=r+r+\frac{\pi d}{3}=9+9+\frac{18 \pi}{3}=18+6 \pi \ cm\end{align*}

#### Practice

1. Find the area and circumference of a circle with radius 3 *in*.

2. Find the area and circumference of a circle with radius 6 *in*.

3. Find the area and circumference of a circle with diameter 15 *cm*.

4. Find the area and circumference of a circle with diameter 22 *in*.

5. Find the area of a circle with a circumference of \begin{align*}32 \pi \ cm\end{align*}.

6. Find the circumference of a circle with area \begin{align*}32 \pi \ cm^2\end{align*}.

7. The sector below is \begin{align*}\frac{1}{2}\end{align*} of a circle. Find the area of the sector.

8. Find the perimeter of the sector in #7.

9. The sector below is \begin{align*}\frac{3}{4}\end{align*} of a circle. Find the area of the sector.

10. Find the perimeter of the sector in #9.

11. The sector below is \begin{align*}\frac{1}{8}\end{align*} of a circle. Find the area of the sector.

12. Find the perimeter of the sector in #11.

13. The sector below is \begin{align*}\frac{2}{3}\end{align*} of a circle. Find the area of the sector.

14. Find the perimeter of the sector in #13.

15. Explain the formula \begin{align*}A=\frac{Cr}{2}\end{align*} where \begin{align*}A\end{align*} is the area of a circle, \begin{align*}C\end{align*} is the circumference of the circle, and \begin{align*}r\end{align*} is the radius of the circle.