Pentagon Area Calculator
Find the area and perimeter of a regular pentagon.
Results update as you type.
How it works
A regular pentagon is a five-sided polygon with all sides and interior angles equal, and this calculator finds its area and perimeter from one side length. You enter the side, and it returns the area using the exact closed-form formula and the perimeter as five times the side. The area formula involves the golden ratio's characteristic square roots, reflecting the deep connection between the pentagon and the golden ratio: the diagonal of a regular pentagon divided by its side equals the golden ratio, about 1.618. Numerically, the area is roughly 1.7205 times the side squared. Each interior angle of a regular pentagon is 108 degrees, and the five angles sum to 540 degrees. Pentagons appear in architecture, the famous Pentagon building, in flowers and starfish, in soccer ball panels, and in decorative tiling, though unlike hexagons they cannot tile a plane on their own. This calculator assumes a regular pentagon with equal sides; an irregular pentagon would need to be split into triangles or handled with coordinate methods. Enter the side length to compute the area and perimeter of your regular pentagon.
Area = (1/4)·√(5(5 + 2√5))·s²
The area of a regular pentagon is one quarter of the square root of five times the quantity five plus twice the square root of five, all multiplied by the side length squared. This constant, roughly 1.7205, times the side squared gives the area. The perimeter is five times the side length because a regular pentagon has five equal sides.
Source:Euclidean geometry
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter side length.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Area updates instantly as you type — no button to press.
Frequently asked questions
How do you find the area of a regular pentagon?
Multiply one quarter of the square root of five times five plus twice the square root of five by the side length squared. This constant is about 1.7205, so the area is roughly 1.7205 times the side squared. For a side of two, the area is about 6.88 square units, which this calculator computes exactly.
What is the interior angle of a regular pentagon?
Each interior angle of a regular pentagon measures exactly 108 degrees, and the five interior angles together sum to 540 degrees. Because 108 does not divide evenly into 360, regular pentagons cannot tile a flat plane by themselves without leaving gaps, unlike triangles, squares, and regular hexagons, which fit together perfectly.
How is the pentagon related to the golden ratio?
In a regular pentagon, the ratio of a diagonal to a side equals the golden ratio, approximately 1.618. The square roots of five appearing in the area formula stem from this relationship. Drawing all the diagonals of a pentagon creates a pentagram, whose segments are also in golden-ratio proportions, a fact prized since antiquity.
Can regular pentagons tile a surface?
No, regular pentagons cannot tile a plane on their own because their 108-degree angles do not combine to make 360 degrees around a point. However, certain irregular pentagons can tile, and pentagons appear in combination with other shapes in many tiling and architectural patterns, as well as on the surface of a soccer ball with hexagons.
Does this work for irregular pentagons?
No. This calculator assumes a regular pentagon with five equal sides and equal angles. For an irregular pentagon whose sides differ, a single side length is not enough. You would divide the shape into triangles and sum their areas, or apply the shoelace formula using the coordinates of the five vertices.