## Related questions with answers

The United State 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.

Solutions

VerifiedThe area of a regular pentagon with side length $s$ is calculated by the formula:

$\boxed {A=\dfrac {1}{4} \cdot \sqrt {5(5+2\cdot \sqrt {5})}\cdot s^2}$

The area of the united states pentagon:

$\begin{align*} A&=\dfrac {1}{4} \cdot \sqrt {5(5+2\cdot \sqrt {5})}\cdot 921^2 \\ &\boxed {\color{#c34632} \approx 1,459,379 \text { ~ft}^2} \end{align*}$

A regular pentagon can be divided into 5 congruent isosceles triangles each having a vertex angle of:

$\dfrac{360\text{\textdegree}}{5}=72\text{\textdegree}$

One isosceles triangle has a base of 921 ft and we can find its area then multiply it by 5 to find the area of the Pentagon. An isosceles triangle can be divided into two $36\text{\textdegree}$-$54\text{\textdegree}$-$90\text{\textdegree}$ triangles along its height, which we name $h$ as shown:

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