How to find a regular pentagon's area
A regular pentagon has five equal sides and angles. Its area is A = (1/4) × √(5(5 + 2√5)) × s², where s is the side length. The constant works out to about 1.7205, so A ≈ 1.7205 × s². For a side of 4, A = about 27.53 square units.
The formula comes from splitting the pentagon into five identical isosceles triangles that meet at the centre. Each has base s and a height equal to the apothem, and summing their areas produces the constant 1.7205 once the trigonometry is simplified. Split the pentagon into five triangles from the centre and the triangle area calculator reaches the same figure.
How to use this calculator
Enter the side length — the length of one of the five equal edges. The result is in square units of that length. All sides of a regular pentagon are equal, so one measurement is enough.
Do not enter the distance across the pentagon or the radius to a corner; both are larger than the side and will overstate the area. Because s is squared, a pentagon with side 8 has four times the area of one with side 4.
Pentagon among polygons
A pentagon is the 5-sided case of the general regular-polygon formula. For other shapes use the regular polygon area calculator, or jump to the six-sided hexagon area calculator.
Each interior angle of a regular pentagon is 108°, and the five angles total 540°. As the number of sides rises, a regular polygon with a fixed side length covers more area and approaches a circle, which is why the pentagon constant of 1.7205 sits between the square's 1 and the hexagon's roughly 2.598.
Frequently asked questions
- What is the formula for a regular pentagon's area?
- A = (1/4) × √(5(5 + 2√5)) × s², where s is the side length. The constant is about 1.7205.
- What is the area of a pentagon with side 4?
- About 27.53 square units.
- Does this work for irregular pentagons?
- No — this formula is for regular pentagons with five equal sides. Irregular shapes need to be split into triangles.
- What units is the area in?
- Square units of whatever length unit you used for the side.
- Can I find the area if I only know the apothem?
- Yes, using the general polygon formula A = ½ × perimeter × apothem. For a regular pentagon that is A = ½ × 5s × a, so you first need the side length s — the apothem alone is enough only if you convert it back to s first.