Question

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

Answer 1

To find the length of a 45° arc in a circle, we can use the formula for the arc length, which is given by:

\[
\text{Arc length} = \frac{\theta}{360} \times 2\pi r
\]

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

In this case, the radius \(r\) is 7 centimeters and the angle \(\theta\) is 45°.

Plugging the values into the formula:

\[
\text{Arc length} = \frac{45}{360} \times 2\pi \times 7
\]

First, simplify \(\frac{45}{360}\):

\[
\frac{45}{360} = \frac{1}{8}
\]

Now substituting this back into the equation:

\[
\text{Arc length} = \frac{1}{8} \times 2\pi \times 7
\]

Calculating \(2\pi \times 7\):

\[
2\pi \times 7 = 14\pi
\]

Now substitute this back into the arc length equation:

\[
\text{Arc length} = \frac{1}{8} \times 14\pi = \frac{14\pi}{8} = \frac{7\pi}{4}
\]

Therefore, the length of the 45° arc is:

\[
\frac{7\pi}{4} \text{ centimeters}
\]

To give a numerical approximation, if we take \(\pi \approx 3.14\):

\[
\text{Arc length} \approx \frac{7 \times 3.14}{4} \approx \frac{21.98}{4} \approx 5.495 \text{ centimeters}
\]

Thus, the exact length of the arc is \(\frac{7\pi}{4} \text{ cm}\) and the approximate length is about \(5.495 \text{ cm}\).