In geometry, students work with a lot of different shapes. One of the most important polygons to become familiar with is the pentagon.

Pentagons – A Few Geometrical Facts

All pentagons have five straight sides, but the sides do not have to be of equal length.

A regular pentagon has five equal sides and five equal angles. In basic geometry, most problems will involve regular polygons.

Each interior angle of a regular pentagon = 108 degrees.

Each exterior angle of a regular pentagon = 72 degrees.

The sum of the interior angles of a regular pentagon = 540 degrees.

Drawing diagonal lines between the points of pentagon will result in a perfect star shape or pentagram.

If the five sides of a shape are NOT connected or the shape has a curved side(s), it is NOT a pentagon.

Types of Pentagons

Regular or equilateral pentagon: five equal sides and angles

Irregular pentagon: five unequal sides and unequal angles

Convex pentagon: no internal angle can be greater than 180 degrees

Concave pentagon: has an internal angle greater than 180 degrees causing two sides to “sink in” like a “cave”

Parts of a Pentagon  

Side: one of the five line segments

Vertex: two sides meet at a point called a vertex

Diagonal: a line that connects two vertices that isn’t one of the five sides

Interior angle: an inside angle formed by two sides of the pentagon

Exterior angle: an angle on the outside of the pentagon formed by two adjacent sides

Pentagon Formulas

Area: an easy formula for finding the approximate area of a regular pentagon is 1.7204774 * s2 (s = the length of the side of the polygon)

n = number of sides (5 for a pentagon)

Sum of the interior angles of a polygon/pentagon = (n-2) * 180 degrees

The number of diagonals in a pentagon =   ½ n(n-3)

Interior angle =

Pentagons – A Few Fun Facts

Why the Pentagon is a pentagon: The headquarters of the U.S. Dept. of Defense in Washington, DC is named the Pentagon. This massive concrete and steel building has a total floor area of almost 7 million square feet and 17.5 miles of corridors. At the beginning of WWII in 1941, President Roosevelt decided a new building was needed for the War Dept.

The architect chose to take advantage of the properties of a symmetrical pentagon. Doing so shortened the distance people would have to walk from one office to another within this huge building when compared to a traditional rectangular building. A circular building would have also incorporated shorter walking distances, but constructing a building with straight sides such as a pentagon was much easier and faster.

Okra: Next time you’re eating fried okra or gumbo, take a look at a slice of okra. It’s shaped like a pentagon.

Starfish: Almost all starfish have a fivefold radial symmetry or are shaped like pentagons.

Poetry: There is actually something known as pentagon poetry.

Musical Pentagons: If you like music from the 1980s, check out Rich Clare’s Pentagon Band. For something different, South Korea has a boy band named Pentagon.

As you can see, a pentagon is a very useful shape. Not only is a pentagon frequently used in basic geometry, it is a shape useful in architecture and found throughout the natural world.