Properties of Circles
[latex-display]\begin{array}{c}r\text{ is the length of the radius.}\hfill \\ d\text{ is the length of the diameter.}\hfill \end{array}[/latex-display]
[latex-display]\begin{array}{cccc}\text{The circumference is }2\pi \mathit{\text{r}}.\hfill & & & C=2\pi \mathit{\text{r}}\hfill \\ \text{The area is }\pi{\mathit{\text{r}}}^{2}.\hfill & & & A=\pi{\mathit{\text{r}}}^{2}\hfill \end{array}[/latex-display]
Since the diameter is twice the radius, another way to find the circumference is to use the formula [latex]C=\pi \mathit{\text{d}}[/latex].
Suppose we want to find the exact area of a circle of radius [latex]10[/latex] inches. To calculate the area, we would evaluate the formula for the area when [latex]r=10[/latex] inches and leave the answer in terms of [latex]\pi[/latex].
[latex-display]\begin{array}{}\\ A=\pi {\mathit{\text{r}}}^{2}\hfill \\ A=\pi \text{(}{10}^{2}\text{)}\hfill \\ A=\pi \cdot 100\hfill \end{array}[/latex-display]
We write [latex]\pi [/latex] after the [latex]100[/latex]. So the exact value of the area is [latex]A=100\pi [/latex] square inches.
To approximate the area, we would substitute [latex]\pi \approx 3.14[/latex].
[latex-display]\begin{array}{ccc}A& =& 100\pi \hfill \\ \\ & \approx & 100\cdot 3.14\hfill \\ & \approx & 314\text{ square inches}\hfill \end{array}[/latex-display]
Remember to use square units, such as square inches, when you calculate the area.
example
A circle has radius [latex]10[/latex] centimeters. Approximate its ⓐ circumference and ⓑ area.
Solution
ⓐ Find the circumference when [latex]r=10[/latex]. |
|
Write the formula for circumference. |
[latex]C=2\pi \mathit{\text{r}}[/latex] |
Substitute [latex]3.14[/latex] for [latex]\pi [/latex] and 10 for , [latex]r[/latex] . |
[latex]C\approx 2\left(3.14\right)\left(10\right)[/latex] |
Multiply. |
[latex]C\approx 62.8\text{ centimeters}[/latex] |
ⓑ Find the area when [latex]r=10[/latex]. |
|
Write the formula for area. |
[latex]A=\pi {\mathit{\text{r}}}^{2}[/latex] |
Substitute [latex]3.14[/latex] for [latex]\pi [/latex] and 10 for [latex]r[/latex] . |
[latex]A\approx \left(3.14\right){\text{(}10\text{)}}^{2}[/latex] |
Multiply. |
[latex]A\approx 314\text{ square centimeters}[/latex] |
example
A circle has radius [latex]42.5[/latex] centimeters. Approximate its ⓐ circumference and ⓑ area.
Answer:
Solution
ⓐ Find the circumference when [latex]r[/latex]=[latex]42.5[/latex]. |
|
Write the formula for circumference. |
[latex]C=2\pi \mathit{\text{r}}[/latex] |
Substitute [latex]3.14[/latex] for [latex]\pi [/latex] and [latex]42.5[/latex] for [latex]r[/latex] |
[latex]C\approx 2\left(3.14\right)\left(42.5\right)[/latex] |
Multiply. |
[latex]C\approx 266.9\text{ centimeters}[/latex] |
ⓑ Find the area when [latex]r=42.5[/latex] . |
|
Write the formula for area. |
[latex]A=\pi {\mathit{\text{r}}}^{2}[/latex] |
Substitute [latex]3.14[/latex] for [latex]\pi [/latex] and [latex]42.5[/latex] for [latex]r[/latex] . |
[latex]A\approx \left(3.14\right){\text{(}42.5\text{)}}^{2}[/latex] |
Multiply. |
[latex]A\approx 5671.625\text{ square centimeters}[/latex] |
example
A circle has radius [latex]\frac{14}{15}[/latex] meter. Approximate its ⓐ circumference and ⓑ area.
Answer:
Solution
ⓐ Find the circumference when [latex]r=\frac{14}{15}[/latex]. |
|
Write the formula for circumference. |
[latex]C=2\pi \mathit{\text{r}}[/latex] |
Substitute [latex]\frac{22}{7}[/latex] for [latex]\pi [/latex] and [latex]\frac{14}{15}[/latex] for [latex]r[/latex] . |
[latex]C\approx 2\left(\frac{22}{7}\right)\left(\frac{14}{15}\right)[/latex] |
Multiply. |
[latex]C\approx \frac{88}{15}\text{meters}[/latex] |
ⓑ Find the area when [latex]r=\frac{14}{15}[/latex]. |
|
Write the formula for area. |
[latex]A=\pi {\mathit{\text{r}}}^{2}[/latex] |
Substitute [latex]\frac{22}{7}[/latex] for [latex]\pi [/latex] and [latex]\frac{14}{15}[/latex] for [latex]r[/latex] . |
[latex]A\approx \left(\frac{22}{7}\right){\text{(}\frac{14}{15}\text{)}}^{2}[/latex] |
Multiply. |
[latex]A\approx \frac{616}{225}\text{square meters}[/latex] |