The pentagonal shape, the idea of architect G. Edwin Bergstrom, was adapted from army forts that were similarly shaped, thus giving the Pentagon the image of a fortress. It consists of five concentric pentagonal rings connected by perpendicular corridors.

Restrictions -
Headquarters big enough to hold 40,000 people,
Parking for 10,000 cars.
4 million square feet of office space
It must be no more than four stories high (it ended up being five stories high) — a tall building would obstruct views of Washington and require too much steel, urgently needed for battleships and weapons.
(Number of floors, plus mezzanine and basement: seven, five above ground, two below ground.)

The restrictions were confounding, given the space they needed. The easiest solution, constructing a tall building, was out. They would have to spread out horizontally. But how? A square building that size — with the enormous interior distances to be covered — was too unwieldy, as was a rectangle. The Arlington Farm tract had a peculiar asymmetrical pentagon shape bound on five sides by roads or other divisions. Finally, guided by the odd shape of the plot, they designed an irregular pentagon. A sketch by Socrates Thomas Stathes, a young War Department draftsman, showed a square with a corner cut off, more or less matching the tract’s shape. It was really two buildings, a five-sided ring surrounding a smaller one of the same shape.
References :
http://www.washingtonpost.com/wp-dyn/content/article/2007/05/23/AR2007052301296_pf.html

Theres a couple of different reasons:

1)Back in the 1400′s some of the most successfull forts in europe were called ‘snowflake forts" that were shaped like a pentagon.

2)The Pentagon, aka the 5 sided trash can, also lovingly refered to as "HINDquarters" (Pun on the word "Headquarters"- as in the HINDquarters on an ass) orginally had one wing for: Army, Navy, Marines, Coast Guard and Army Air Corp- 5 military branches, required a building with 5 sides.

3)There is speculation with physical evidence that Washington, DC may have been designed and built by Masons- this would also include the Pentagon.

Fort Pulaski was probably designed by some Major General who wanted to be assigned to the Pentagon to advance his career and since they wouldn’t send him, he built it to make his own pentagon. Stranger things have happened- thats why NCO’s train the officers
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I am a former army drill sgt

Two different reasons. Many of the pre-civil war forts were designed as stars and pentagons, but this design was purely for military reasons than a cool design feature. With a star or pentagon shape, the troops garrisoned at the forts would be able to have greater fields of fire than they would if the shape was any other shape as well as having a limited distance for troops to move across on the interior in case on part or another needed reinforcing. Having a greater field of fire means, in essence, that you could employ more cannons or rifle on an approaching enemy. Unfortunately this is only true to an extent. When the enemy got suffciently close, defenders would not be able to fire on them at all because they would lose them from view. To counter this small design flaw, additional extensions were placed at each point of the pentagon, forming it into more a star configuration. This allowed defenders to fire down the oustide walls to pick off enemy that got close in. Pulaski, Sumter, McHenry, and numerous other have this design.

The pentagon was designed with some limitations. First it had to be big, and it couldn’t be tall (Didn’t want to block the views of DC and shortgaes of steel prevented it also). It could have been built in a square of rectangular fashion, but that sort of design would have made it difficult for a person to walk from one place to another. So in the end a pentagon shaped design was created. it met the space requirements, and kept distances short.

What your math teacher was probably trying to get you to understand is this. With a pentagon shape, like a circle creates shorter walking distances on the interior than a similar sized square or rectangular building.
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