Exercise 53 Page 695

We want to find the area of a regular hexagon with a perimeter of $36\text{ in}.$ To do so, we will start by finding its side length and the measure of the angles formed by two radii. Then, we will use a right triangle to find the apothem. Finally, we will use the formula $A=\frac{1}{2}ap$ to find the area of the polygon.

Length of a Side

Let's start by drawing our regular hexagon.

All $6$ sides are congruent, and we are told that the perimeter is $36\text{ in}.$ Therefore, we can find the length of a side by dividing $36$ by $6.$ The length of each side is $6\text{ in}.$

Measure of the Angles Formed by Two Radii

Now let's draw the radii of the polygon. Since all of the radii are congruent, they form $6$ congruent isosceles triangles. Moreover, since corresponding angles of congruent figures are congruent, all the vertex angles of the isosceles triangles formed by the radii are congruent.

Recall that a full turn of a circle measures $360^{\circ }.$ Therefore, we can find the measure of the central angles by dividing $360$ by the number of angles. The measure of the vertex angle of each of the isosceles triangles formed by the radii is $60^{\circ }.$

Apothem

Let's consider one of the isosceles triangles formed by two radii and a side of the hexagon.

The apothem bisects both the angle whose measure is $60^{\circ }$ and the side whose length is $6\text{ in}.$ Therefore, the apothem divides the isosceles triangle into two right triangles with an acute angle of measure $\frac{60^{\circ }}{2}=30^{\circ }$ and opposite side length $\frac{6}{2}=3\text{ in}.$ Let's look at just one of these right triangles.

In the above right triangle we know the measure of an $\text{angle}$ and the length of its $\text{opposite}$ $\text{side},$ so we can use the tangent ratio to find the length of its $\text{adjacent}$ $\text{side}.$ We found that the apothem of the regular polygon is $3\sqrt{3}\text{ in}.$

Area of the Regular Polygon

The area of a regular polygon is half the product of the apothem and the perimeter. Since we already know that the apothem is $3\sqrt{3}\text{ in.}$ and the perimeter is $36\text{ in.},$ we can substitute these values in the formula $A=\frac{1}{2}ap$ to find the area. The area of the regular polygon to the nearest tenth is $54\sqrt{3}{\text{ in.}}^2.$ Therefore, the correct answer is D.