Chapter 9.10, Problem 14PSB

To determine

To find:The length of the apothem of a regular pentagon

Answer to Problem 14PSB

The length of the apothem is $6.882\text{ cm}$

Explanation of Solution

Given information:The following information has been given

Perimeter of the regular pentagon is $50\text{ cm}$

Formula used:Basic tangent definition has been used

Calculation:We know that as the perimeter is $50\text{ cm}$ , hence each side will be of $10cm$

Now, we consider an apothem from the center of the pentagon to one of its sides. Hence, we can say that

  $\tan \frac{A}{2}=\frac{x}{5\text{ cm}}$

Where, A is the interior angle of each vertex of the polygon, and x is the measure of the apothem. Hence, we can say that

  $\tan \frac{{108}^0}{2}=\tan {54}^0=1.3764$

Also, we can say that

  $\tan {54}^0=\frac{x}{5\text{ cm}}$

Hence, from the above equation, we can say that

  $1.3764=\frac{x}{5\text{ cm}}$

Thus, we get

  $x=6.882\text{ cm}$