Chapter 11, Problem 1PST

To determine

To Find: The area of the figure.

Answer to Problem 1PST

The correct option is A.

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the given figure is the regular hexagon then the apothem is,

  $\begin{array}{c}a=\frac{20}{2}\\ =10\text{in}\end{array}$

Consider the formula for the area of the polygon with the apothem is,

  $S=n{a}^2\left(\frac{180°}{n}\right)$

Consider the figure shown in Figure 2

  

Figure 2

Consider by the definition of the tangent is,

  $\begin{array}{l}\tan \left(\frac{180°}{n}\right)=\frac{\left(\frac{s}{2}\right)}{a}\\ a\tan \left(\frac{180°}{n}\right)=\frac{s}{2}\\ 2a\tan \left(\frac{180°}{n}\right)=s\end{array}$

Then, the area of the given figure is,

  $\begin{array}{c}A=\frac{1}{2}ap\\ =\frac{1}{2}a\left(n\left(2a\tan \left(\frac{180°}{n}\right)\right)\right)\\ =n{a}^2\tan \left(\frac{180°}{n}\right)\end{array}$

Then, the required area is calculated as,

  $\begin{array}{c}S=6{\left(10\right)}^2\tan \left(\frac{180°}{5}\right)\\ =346.4{\text{in}}^2\end{array}$