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

7 Things You Might Not Know About Pentagons

  1. All pentagons have five straight sides, but the sides do not have to be of equal length.
  2. A regular pentagon has five equal sides and five equal angles. In basic geometry, most problems will involve regular polygons.
  3. Each interior angle of a regular pentagon = 108 degrees.
  4. Each exterior angle of a regular pentagon = 72 degrees.
  5. The sum of the interior angles of a regular pentagon = 540 degrees.
  6. Drawing diagonal lines between the points of pentagon will result in a perfect star shape or pentagram.
  7. 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

  1. Regular or equilateral pentagon: five equal sides and angles
  2. Irregular pentagon: five unequal sides and unequal angles
  3. Convex pentagon: no internal angle can be greater than 180 degrees
  4. Concave pentagon: has an internal angle greater than 180 degrees causing two sides to “sink in” like a “cave”

Parts of a Pentagon  

  1. Side: one of the five line segments
  2. Vertex: two sides meet at a point called a vertex
  3. Diagonal: a line that connects two vertices that isn’t one of the five sides
  4. Interior angle: an inside angle formed by two sides of the pentagon
  5. Exterior angle: an angle on the outside of the pentagon formed by two adjacent sides

How to Calculate the Area of a Pentagon

  1. Start with one side and the apothem*
  2. Divide the pentagon into 5 triangles by drawing 5 lines from the center of the pentagon
  3. Calculate the area of a triangle**
  4. Multiply by 5 to find the total area

*The apothem is the line from the center of the pentagon to a side, intersecting the side at a 90º right angle.

**Remember the formula for calculating the area of a triangle is ½ x base x height

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.