Question

the polygon below is a regular hexagon. what is the approximate length of the apothem? round to the nearest tenth 9.0 cm 15.6 cm 20.1 cm 25.5 cm 18 cm 18 cm

Explanation:

Step1: Find the central angle

For a regular hexagon, the central angle is $\frac{360^{\circ}}{6}=60^{\circ}$. Half of this central angle is $\frac{60^{\circ}}{2}=30^{\circ}$.

Step2: Use trigonometric ratio

We know that in a right - triangle formed (with the radius as the hypotenuse \(r = 18\) cm and the apothem \(a\) as one of the legs), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\) and the adjacent side is the apothem \(a\), hypotenuse \(r=18\) cm.
So \(a = r\cos\theta\).
Substitute \(r = 18\) and \(\theta=30^{\circ}\), we get \(a=18\times\cos30^{\circ}\).
Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\), then \(a = 18\times0.866=15.588\approx15.6\)

Answer:

\(15.6\) cm