What if you wanted to figure out the area of a circle with a radius of 5 inches?
Area of a Circle
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*}.
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.
Finding the Area
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*}.
Finding the Radius
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.
Calculating the Area of a Circle Inscribed in a Square
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*}
Earlier Problem Revisited
A circle with a radius of 5 inches has area \begin{align*}\pi5^2 = 25\pi \ in^2 \end{align*}.
Examples
Example 1
Find the area of the shaded region from Example C.
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*}
Example 2
Find the diameter of a circle with area \begin{align*}36 \pi\end{align*}.
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.
Example 3
Find the area of a circle with diameter 20 inches.
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*} .
Interactive Practice
Review
Fill in the following table. Leave all answers in terms of \begin{align*}\pi\end{align*}.
radius | area | diameter | |
---|---|---|---|
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*} |
Find the area of the shaded region. Round your answer to the nearest hundredth.
- 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?
Review (Answers)
To view the Review answers, open this PDF file and look for section 10.10.