Chapter 5.6, Problem 30E

To determine

To find: The area of the courtyard.

Answer to Problem 30E

  $213,940{\text{ ft}}^2$

Explanation of Solution

Given information: Architecture: The center of the Pentagon in Arlington, Virginia, is a courtyard in the shape of a regular pentagon. The pentagon could be inscribed in a circle with radius of 300 feet.

Calculation:

This is the part of the pentagon we are interested to get out numbers. If the pentagon is inscribed in a circle (or the circle circumscribed around the pentagon, same effect), the radius of the circle is the radius of the pentagon which is either of the two line segments drawn from the center. We are going to pull out the dark triangle and then look at the right half of it. The central angle of the dark triangle (the center angle) is $=\frac{360}{5}={72}^{\circ }$ half of this will be ${36}^{\circ }$ the radius is given as 300 feet.

I have labeled some parts; we are going to draw the triangle to the right of the apothem.

We want the area. The area formula for a regular polygon is $A=\frac{1}{2}aP$ where a is the apothem andP is the perimeter. This means we need the apothem and the ½ side length from our figure,

  $\begin{array}{l}\cos {36}^{\circ }=\frac{\text{apothem}}{300}\\ \text{apothem}=\cos {36}^{\circ }\times 300=242.7\text{ feet}\\ {\text{sin36}}^{\circ }=\frac{1/2\text{ side}}{300}\\ 1/2\text{ side}=\sin {36}^{\circ }\times 300=176.3\text{ feet}\end{array}$

We need to double this to get the full side of the pentagon.

  $\begin{array}{l}176.3\times 2=352.6\\ \text{Perimeter}=352.6\times 5=1763\text{ feet}\end{array}$

  $A=\frac{1}{2}aP\text{ becomes }A=\frac{1}{2}\times 242.7\times 1763=213,940$

Thus, $\text{Area}=213,940{\text{ ft}}^2$