Chapter 11.5, Problem 5PSA

a

To determine

To calculate : The length of a side of the regular hexagon with radius equal to $12$

Answer to Problem 5PSA

The length is $a=12$

Explanation of Solution

Given information : The radius of the regular hexagon is, $r=12$

Formula used : The side length of a regular hexagon $(a)=2(r\cos {60}^0)$

Calculation : Substituting the value of $r=12$ in the equation, we get

  $a=2(12\cos {60}^0)=12$

Thus, we get

  $a=12$

b

To determine

To calculate : The apothem of the regular hexagon with side length equal to $12$

Answer to Problem 5PSA

The apothem is $h=6\sqrt{3}$

Explanation of Solution

Given information : The side of the regular hexagon is, $a=12$

Formula used : Theapothem of a regular hexagon $(h)=\frac{1}{2}\left(a\tan {60}^0\right)$

Calculation : Substituting the value of $a=12$ in the equation, we get

  $h=\frac{1}{2}\left(12\tan {60}^0\right)=6\sqrt{3}$

Thus, we get

  $h=6\sqrt{3}$

c

To determine

To calculate : The area of the regular hexagon with side length $12$

Answer to Problem 5PSA

The area is $A=216\sqrt{3}$

Explanation of Solution

Given information : The side of the regular hexagon is, $a=12$

Formula used : We use the following formula to find the area of an regular hexagon

  $A=\frac{3\sqrt{3}}{2}{a}^2$

Calculation : Substituting the value of $a=12$ in the equation, we get

  $\begin{array}{l}$ A=\frac{3\sqrt{3}}{2}{(12)}^2$A=216\sqrt{3}\end{array}$

Thus, we get

  $A=216\sqrt{3}$