How do you calculate the area of a pentagon?

There are many different types of pentagons, and many different ways to calculate the area.

The General Case

All pentagons can be subdivided into three triangles by drawing two line segments between pairs of pentagon vertices. There are three conditions that must be met when creating the two lines:
  1. The lines must not cross.
  2. The lines must not connect adjacent verticies.
  3. The lines must be contained completely inside the pentagon.
Once the three triangle are determined then the area of the pentagon can be calculated as the sum of the areas of the three triangles.

The Regular Pentagon

A regular pentagon has five sides of the same length. Also, the interior angles are equivalent In this case the area of the pentagon can be calculated from the side length.

The approximately correct formula (s = side length):
A = s * s * 1.720477401. Put the word 'approximately' into context. If the length of a side is accurate to 2 decimal places, then using the factor of 1.72 will be accurate enough for you; using the 9 decimal places will not increase the accuracy of your answer beyond 2 decimal places. If you know the length of a side to an accuracy of 9 decimal places, the above factor will do the trick. But how often do you need accuracy to that degree? The 'exactly correct' solution below is exactly correct only in theory. When you plug real numbers in, your answer can be no closer to correct than the numbers you enter. So save yourself a lot of trouble and use the factor above.

The exactly correct formula (s = side length, sqrt means square root):

A = s2 * 1/4 sqrt [(25 +10 (sqrt5)]

The formula is derived at the link below.