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 14.1 mm 14.1 mm 22.8 mm 22.8 mm 12.0 mm

Answer 1

In a regular pentagon, the relationship between the radius (R) of the circumscribed circle and the side length (s) can be calculated using the formula:

\[ s = R \times 2 \times \sin\left(\frac{\pi}{n}\right) \]

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

Here, the radius \(R = 12\) mm. Plugging in the values:

\[ s = 12 \times 2 \times \sin\left(\frac{\pi}{5}\right) \]

Calculating \(\sin\left(\frac{\pi}{5}\right)\):

\[ \sin\left(\frac{\pi}{5}\right) \approx 0.5878 \]

Now, substituting back:

\[ s = 12 \times 2 \times 0.5878 \approx 12 \times 1.1756 \approx 14.1 \text{ mm} \]

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

The closest provided option is 14.1 mm.