Regular Pentagon Area Calculator
Calculate the area of a regular pentagon from its side length. Formula: Area = (s^2 / 4) x sqrt(5 x (5 + 2 x sqrt(5))). Instant result.
How to use this tool
- Enter side length (s) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your area and the full breakdown beneath it.
Formula
Area = (s^2 / 4) x sqrt(5 x (5 + 2 x sqrt(5)))
How it works
A regular pentagon has five equal sides and five equal interior angles of 108 degrees. Its area can be computed using the formula with the constant sqrt(5*(5+2*sqrt(5))) / 4, which equals approximately 1.7205 times the side squared.
Worked example
Regular pentagon with side 5 m
- Constant K = sqrt(5 x (5 + 2 x sqrt(5))) / 4 = 6.8819 / 4 = 1.7205
- Area = K x s^2 = 1.7205 x 25 = 43.0119 m^2
43.0119 m^2
Common mistakes to avoid
- Confusing the side length with the apothem or the diagonal; only the side length s goes into the standard formula.
- Using an imprecise value for the nested square roots in sqrt(5*(5+2*sqrt(5))), which should be computed with full precision.
- Applying this formula to an irregular pentagon; it is valid only for a regular pentagon.
Key terms
- What is a regular pentagon?
- A five-sided polygon with all sides equal and all interior angles equal to 108 degrees.
Frequently asked questions
- What is a regular pentagon?
- A regular pentagon has five equal sides and five equal interior angles of 108 degrees each.
- How is the pentagon area formula derived?
- The formula comes from dividing the pentagon into 5 isosceles triangles meeting at the center and computing each triangle's area using the central angle of 72 degrees.
- If I know the perimeter instead of the side, how do I use this calculator?
- Divide the perimeter by 5 to get the side length s, then enter that value.