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.

Answer 1

To find the length of each side of a regular pentagon given the radius (circumradius) \( R \), we can use the formula for the side length \( s \) of a regular pentagon:

\[
s = R \cdot \sqrt{2(1 - \cos(\frac{2\pi}{5}))}
\]

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

\[
\frac{2\pi}{5} \approx 1.2566 \text{ radians}
\]

Using a calculator or trigonometric tables, we find:

\[
\cos(\frac{2\pi}{5}) \approx 0.3090
\]

Now, substitute this into the formula:

\[
1 - \cos\left(\frac{2\pi}{5}\right) \approx 1 - 0.3090 = 0.6910
\]

Next, we find \( 2(1 - \cos(\frac{2\pi}{5})) \):

\[
2(1 - \cos(\frac{2\pi}{5})) \approx 2 \times 0.6910 = 1.3820
\]

Then we take the square root:

\[
\sqrt{1.3820} \approx 1.1750
\]

Now, use the circumradius \( R = 12 \) mm to find the side length \( s \):

\[
s \approx 12 \cdot 1.1750 \approx 14.1
\]

Therefore, the length of each side of the pentagon, rounded to the nearest tenth, is:

\[
\boxed{14.1} \text{ millimeters}
\]