Question

A regular pentagon has a radius of 12 millimeters. What is the measure of each side of the pentagon? Round the answer to the nearest tenth.(1 point)
Responses

12.0 mm
12.0 mm

14.1 mm
14.1 mm

19.4 mm
19.4 mm

22.8 mm

Answer 1

To find the length of each side of a regular pentagon inscribed in a circle (circumcircle) with a given radius, you can use the following formula:

\[
s = r \cdot \sqrt{2(1 - \cos(\frac{2\pi}{n}))}
\]

where \(s\) is the length of each side, \(r\) is the radius, and \(n\) is the number of sides (which is 5 for a pentagon).

Let's calculate it for a pentagon with a radius of 12 mm:

1. **Find \(\cos(\frac{2\pi}{5})\):**

\[
\frac{2\pi}{5} \approx 1.25664
\]
\[
\cos(\frac{2\pi}{5}) \approx 0.309
\]

2. **Substituting values into the side length formula**:

\[
s = 12 \cdot \sqrt{2(1 - 0.309)}
\]
\[
= 12 \cdot \sqrt{2(0.691)}
\]
\[
= 12 \cdot \sqrt{1.382}
\]
\[
\approx 12 \cdot 1.176
\]
\[
\approx 14.112
\]

Rounding to the nearest tenth gives us approximately:

\[
\text{Side length} \approx 14.1 \text{ mm}
\]

Therefore, the answer is **14.1 mm**.