Chapter 5.6, Problem 47E

Solution

To find: The correct option from the given options.

The correct option is (B).

Given information:

The radius of the circle is $12\text{units}$ and the number of sides in the polygon are $12$ .

Formula used:

Area of a triangle is given as:

  $A_{\text{triangle }}\text{ =}\frac{1}{2}\times \left(\text{product of two sides}\right)\times \left(\text{sine of the included angle}\right)$

Area of dodecagon is given as:

  $A_{\text{dodecagon }}=12\times A_{\text{dodecagon }}$

Calculation:

A regular hexagon can be divided into $12$ isosceles triangles.

The vertex angle of each triangle can be given as $\frac{{360}^{°}}{12}={30}^{°}$ .

The side length of each triangle is equal to the radius of the circle.

The area of one isosceles triangle can be calculated as $\text{Area =}\frac{1}{2}\times \left(\text{product of two sides}\right)\times \left(\text{sine of the included angle}\right)$

Substitute $12$ for each side of the triangle and $30°$ for included angle.

  $\begin{array}{l}{A}_{\text{triangle }}=\frac{1}{2}(12)(12)\sin {30}^{°}\\ A_{\text{triangle }}=36{\text{units}}^{\text{2}}\end{array}$

Substitute $36$ for area the triangle in $A_{\text{dodecagon }}=12\times A_{\text{dodecagon }}$ to calculate area of dodecagon

  $\begin{array}{c}{A}_{\text{dodecagon }}=12\times 36\\ =432{\text{units}}^{\text{2}}\end{array}$

Therefore, the correct answer is choice (B).