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.


19.4 mm
14.1 mm
12.0 mm
22.8 mm

Answer 1

To find the length of each side of a regular pentagon given the radius (circumradius, or the distance from the center to a vertex), we can use the following formula:

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

where \(s\) is the length of each side, \(R\) is the circumradius, and \(n\) is the number of sides. For a pentagon, \(n = 5\).

Given:
- \(R = 12\) mm
- \(n = 5\)

First, calculate \(\frac{2\pi}{5}\):

\[
\frac{2\pi}{5} \approx 1.256637061
\]

Next, compute \(\cos\left(\frac{2\pi}{5}\right)\):

\[
\cos\left(\frac{2\pi}{5}\right) \approx 0.309016994
\]

Now, use this value in our formula:

\[
s = 12 \cdot \sqrt{2 - 2 \cdot 0.309016994}
\]
\[
= 12 \cdot \sqrt{2 - 0.618033988}
\]
\[
= 12 \cdot \sqrt{1.381966012}
\]
\[
\approx 12 \cdot 1.175570504
\]
\[
\approx 14.10684605
\]

Now, round the result to the nearest tenth:

\[
s \approx 14.1 \text{ mm}
\]

Thus, the measure of each side of the pentagon is approximately **14.1 mm**.