Question

The radius of a circle is 7 feet. What is the length of a 45° arc?

Answer 1

To find the length of a \( 45^\circ \) arc of a circle with a radius of \( 7 \) feet, we can use the formula for the arc length \( L \):

\[
L = r \theta
\]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the angle in radians.

First, we need to convert \( 45^\circ \) to radians. The conversion from degrees to radians is given by:

\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]

So, for \( 45^\circ \):

\[
\theta = 45^\circ \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}
\]

Now we can calculate the length of the arc:

\[
L = r \theta = 7 \text{ feet} \times \frac{\pi}{4}
\]

Calculating this gives:

\[
L = \frac{7\pi}{4} \text{ feet}
\]

To get a numerical estimate, we can use \( \pi \approx 3.14 \):

\[
L \approx \frac{7 \times 3.14}{4} \approx \frac{21.98}{4} \approx 5.495 \text{ feet}
\]

Thus, the length of the \( 45^\circ \) arc is:

\[
\frac{7\pi}{4} \text{ feet} \quad \text{or approximately } 5.495 \text{ feet.}
\]