Chapter 4.8, Problem 45E

Solution

To find: The length of the apothem.

The length of the apothem is approximately 5.19 cm.

Given information:

Given, in a regular polygon all sides have equal length and all angles have equal measure. The required polygon is a regular seven-sided polygon whose sides are 5 cm.

  

Calculation:

The central angle in a polygon is 360°. This central angle is divided into seven equal parts by the sides i.e., 360/7. Further the apothem divides that angle into half i.e., 360/14=25.71°

Also the apothem divides the sides into two halves i.e., 5/2=2.5 cm

The figure representing the data is:

  

Using the trigonometric function definitions,

  $\begin{array}{l}\tan A=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan \left(25.71°\right)=\frac{2.5}{a}\\ a=\frac{2.5}{\tan \left(25.71°\right)}\\ a=\frac{2.5}{0.481}\\ a\approx 5.19\end{array}$

The length of the apothem is approximately 5.19 cm.

Conclusion:

The length of the apothem is approximately 5.19 cm.