What if you were given the radius or diameter of a circle? How could you find the amount of space the circle takes up? After completing this Concept, you'll be able to use the formula for the area of a circle to solve problems like this.
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Guidance
To find the area of a circle, all you need to know is its radius. If \begin{align*}r\end{align*} is the radius of a circle, then its area is \begin{align*}A=\pi r^2\end{align*}.
We will leave our answers in terms of \begin{align*}\pi\end{align*}, unless otherwise specified. To see a derivation of this formula, see http://www.rkm.com.au/ANIMATIONS/animation-Circle-Area-Derivation.html, by Russell Knightley.
Example A
Find the area of a circle with a diameter of 12 cm.
If \begin{align*}d = 12 \ cm\end{align*}, then \begin{align*}r = 6 \ cm\end{align*}. The area is \begin{align*}A=\pi \left(6^2 \right)=36 \pi \ cm^2\end{align*}.
Example B
If the area of a circle is \begin{align*}20 \pi\end{align*} units, what is the radius?
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} units\end{align*}
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 5 cm.
\begin{align*}A=\pi 5^2=25 \pi \ cm^2\end{align*}
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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.
Answers:
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*} .
Explore More
Fill in the following table. Leave all answers in terms of \begin{align*}\pi\end{align*}.
radius | Area | circumference | |
---|---|---|---|
1. | 2 | ||
2. | \begin{align*}16 \pi\end{align*} | ||
3. | \begin{align*}10\pi\end{align*} | ||
4. | \begin{align*}24\pi\end{align*} | ||
5. | 9 | ||
6. | \begin{align*}90\pi\end{align*} | ||
7. | \begin{align*}35\pi\end{align*} | ||
8. | \begin{align*}\frac{7}{\pi}\end{align*} | ||
9. | 60 | ||
10. | 36 |
Find the area of the shaded region. Round your answer to the nearest hundredth.