Question

How do I calculate the side length of a regular pentagon with a radius of 12 millimeters?

Answer 1

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

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

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

Given that the radius \( R = 12 \) millimeters, we can calculate the side length as follows:

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

\[
\frac{2\pi}{5} \approx 1.25664 \text{ radians}
\]
Using a calculator, find \( \cos(1.25664) \):
\[
\cos\left(\frac{2\pi}{5}\right) \approx 0.309
\]

2. Substitute this value into the formula:

\[
s = 12 \cdot \sqrt{2(1 - 0.309)}
\]
First, calculate \( 1 - 0.309 \):
\[
1 - 0.309 \approx 0.691
\]

Then, calculate \( 2 \cdot 0.691 \):
\[
2 \cdot 0.691 \approx 1.382
\]

Now, take the square root:
\[
\sqrt{1.382} \approx 1.176
\]

3. Finally, calculate the side length \( s \):
\[
s \approx 12 \cdot 1.176 \approx 14.112 \text{ mm}
\]

Thus, the side length of the regular pentagon with a circumradius of 12 millimeters is approximately \( 14.11 \) millimeters.