Question
A regular hexagon has side lengths of 8 centimeters. What is the exact measure of its apothem?
Answers
GPT-4o mini
To find the apothem of a regular hexagon with a side length of \( s = 8 \) cm, we can use the formula for the apothem \( a \) of a regular polygon, which is:
\[
a = \frac{s}{2 \tan(\frac{\pi}{n})}
\]
where \( n \) is the number of sides of the polygon. For a regular hexagon, \( n = 6 \).
First, we calculate \( \frac{\pi}{n} \):
\[
\frac{\pi}{6} = \frac{3.14159}{6} \approx 0.5236 \text{ radians}
\]
Next, we find \( \tan\left(\frac{\pi}{6}\right) \):
\[
\tan\left(\frac{\pi}{6}\right) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.5774
\]
Now we can substitute into the apothem formula:
\[
a = \frac{8}{2 \tan\left(\frac{\pi}{6}\right)} = \frac{8}{2 \cdot \frac{1}{\sqrt{3}}} = \frac{8}{\frac{2}{\sqrt{3}}} = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3}
\]
Thus, the exact measure of the apothem of the regular hexagon is:
\[
\boxed{4\sqrt{3}} \text{ centimeters}
\]
\[
a = \frac{s}{2 \tan(\frac{\pi}{n})}
\]
where \( n \) is the number of sides of the polygon. For a regular hexagon, \( n = 6 \).
First, we calculate \( \frac{\pi}{n} \):
\[
\frac{\pi}{6} = \frac{3.14159}{6} \approx 0.5236 \text{ radians}
\]
Next, we find \( \tan\left(\frac{\pi}{6}\right) \):
\[
\tan\left(\frac{\pi}{6}\right) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.5774
\]
Now we can substitute into the apothem formula:
\[
a = \frac{8}{2 \tan\left(\frac{\pi}{6}\right)} = \frac{8}{2 \cdot \frac{1}{\sqrt{3}}} = \frac{8}{\frac{2}{\sqrt{3}}} = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3}
\]
Thus, the exact measure of the apothem of the regular hexagon is:
\[
\boxed{4\sqrt{3}} \text{ centimeters}
\]