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12-57.

The United States Department of Defense is located in a building called the Pentagon because it is in the shape of a regular pentagon. Known as “the largest office building in the world,” its exterior edges measure $921$ feet. Find the area of land enclosed by the outer walls of the Pentagon building.

Step 1:

Draw a sketch of the situation:

A regular pentagon with a side 921 feet.

The pentagon is divided into 5 equal triangles from the center point.

Added to the divided pentagon, From the center point, a perpendicular segment to one of the sides.

Step 2:

Divide the pentagon into 5 equal isosceles triangles:

Step 3:

Draw a height for one of the triangles, dividing it into two right triangles:

Step 4:

Find the area using trigonometric ratios.

The legs of a right triangle are 460.5 ft and h. The angle opposite leg, h, is theta.

Hint:

The interior angles of a regular pentagon measure $\frac{180(5-2)}{5}=\frac{180\cdot3}{5}=\frac{540}{5}=\ 108º$

More help:

$\theta\ =\frac{108º}{2}$

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