Pentagon Calculator
Find a regular pentagon's area, perimeter, and diagonal from its side length.
Area
43.0119
Area = (s²/4) × √(5(5 + 2√5)). A side of 5 covers about 43.01 square units.
Perimeter (5s)
25
Diagonal (φ·s)
8.0902
Apothem
3.441
Circumradius
4.2533
How it works
A regular pentagon has five equal sides and five equal angles. That symmetry means one measurement — the side length — locks in every other value, so you never need to measure the diagonals or the height separately.
The area follows the formula (s²/4) × √(5(5 + 2√5)), and the perimeter is simply five times the side. A neat quirk of the pentagon is that its diagonal equals the golden ratio, about 1.618, times the side. The apothem (center to a side) and circumradius (center to a corner) round out the picture.
You'll recognize this shape from home plate on a baseball diamond, the Pentagon building, and plenty of tiling patterns. Enter a side of 5 and the area comes out to roughly 43.01 square units.
Frequently asked questions
How do I find the area of a regular pentagon?
Use area = (s²/4) × √(5(5 + 2√5)), where s is the side length. It's equivalent to splitting the pentagon into five identical triangles that meet at the center and summing their areas.
Why does the diagonal involve the golden ratio?
In a regular pentagon, the ratio of a diagonal to a side is exactly the golden ratio φ ≈ 1.618. So the diagonal is φ × s. It falls straight out of the pentagon's geometry and shows up in the pentagram drawn from its diagonals.
Does this handle irregular pentagons?
No. These formulas assume a regular pentagon with equal sides and angles. An irregular five-sided shape has to be broken into triangles and measured piece by piece.