Pentagon Area
A = (5·l²)/(4·tan(π/5))P = 5·lThe area of a regular pentagon is five times the side squared, divided by four times the tangent of π over five. The resulting factor is roughly 1.721.
The perimeter is five times the side.
The formula requires a **regular** pentagon, with all five sides and angles equal. Each interior angle measures 108 degrees.
Regular triangles, squares and hexagons fill the plane with no gaps. Regular pentagons cannot.
The reason is arithmetic: to close around a vertex, the angles must sum to exactly 360 degrees. At 108 degrees, three pentagons give 324 and leave a 36 degree gap; four would give 432 and overlap.
That is why no floor is tiled with regular pentagons — and why the hexagon, at 120 degrees, fits exactly three times.
The ratio between the diagonal and the side of a regular pentagon is exactly the golden ratio, roughly 1.618.
Drawing all the diagonals produces the pentagram, and every crossing divides the segments in that same ratio. This recurrence fascinated the Pythagoreans and is why the symbol appears throughout historical iconography.
It is also one of the few places where the golden ratio appears by geometric necessity rather than later aesthetic interpretation.
Frequently asked questions
No. The formula assumes five equal sides and angles. For irregular ones, split the figure into triangles and add the areas.
108 degrees. The interior angles of any pentagon sum to 540 degrees, and in a regular one they divide equally among the five.
Because 108 degrees does not divide 360. Three pentagons leave a 36 degree gap at each vertex, and four overlap.
The ratio of the diagonal to the side is exactly the golden ratio. Drawing the diagonals, each crossing divides the segments in that same ratio.