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

19.4 mm
19.4 mm

12.0 mm
12.0 mm

14.1 mm
14.1 mm

22.8 mm

Answer 1

To find the length of each side of a regular pentagon given its radius (the distance from the center to the vertices), we can use the formula for the side length \( s \) of a regular pentagon inscribed in a circle of radius \( r \):

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

where \( n \) is the number of sides (which is 5 for a pentagon).

Using the radius \( r = 12 \) mm and \( n = 5 \):

1. Calculate \( \frac{2\pi}{5} \):
\[
\frac{2\pi}{5} \approx 1.2566 \text{ radians}
\]

2. Compute \( \cos\left(\frac{2\pi}{5}\right) \):
\[
\cos\left(\frac{2\pi}{5}\right) \approx 0.309
\]

3. Substitute this into the formula:
\[
s = 12 \cdot \sqrt{2 - 2 \cdot 0.309}
\]
\[
= 12 \cdot \sqrt{2 - 0.618} = 12 \cdot \sqrt{1.382}
\]
\[
\approx 12 \cdot 1.175 \approx 14.1 \text{ mm}
\]

Thus, the length of each side of the regular pentagon is approximately \( 14.1 \) mm, rounded to the nearest tenth.

The correct answer is:

**14.1 mm**