Regular Polygon Area Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/9/2026
A regular polygon's area is calculated with (number of sides × side length²) ÷ (4 × tan(π ÷ number of sides)). A regular hexagon with a 5 cm side thus has an area of about 64.95 cm².
Explanation
This general formula applies to any regular polygon — a polygon where all sides and all angles are equal — whether a pentagon (5 sides), a hexagon (6 sides), an octagon (8 sides), or any number of sides starting from 3. The underlying geometric principle is to split the polygon into n identical isosceles triangles, each with its apex at the polygon's center, whose area is calculated from the trigonometry of the central angle — the general formula is simply the simplified result of this sum of n triangles. A useful check on this formula: applied to a 4-sided polygon (a square), it gives back exactly side², the familiar area of a square, confirming it's consistent with simpler area formulas for special cases. For a triangle (3 sides) or a circle (the limit of a regular polygon as the number of sides tends to infinity), see instead the triangle area calculator or the circle area calculator, whose dedicated formulas are simpler to use than this general case.
Example: a regular hexagon with a 5 cm side
Inputs
Number of sides: 6. Side length: 5 cm.
Calculation
Area = (6 × 5²) ÷ (4 × tan(π ÷ 6)) = 150 ÷ (4 × 0.57735) = 150 ÷ 2.3094 ≈ 64.95 cm².
Result
A regular hexagon with a 5 cm side has an area of about 64.95 cm².
Frequently asked questions
Does this formula work for an equilateral triangle?
Yes, an equilateral triangle is a regular polygon with 3 sides: entering 3 as the number of sides, the formula gives the same area as a calculation dedicated to the equilateral triangle. For an arbitrary triangle (not necessarily equilateral), use the triangle area calculator instead, which doesn't assume equal sides.
Why does this formula use the tangent function?
Because the regular polygon breaks down into n identical isosceles triangles starting from the center, and the tangent of each triangle's central angle (π ÷ n radians) directly relates the side length to the apothem (the distance from the center to the midpoint of a side), a quantity needed to calculate each triangle's area.
What happens as the number of sides becomes very large?
As the number of sides increases, the regular polygon visually gets closer to a circle, and the area calculated by this formula approaches that of a circle whose circumference would equal the polygon's perimeter. This is a classic illustration of how a circle can be seen as the limit of a regular polygon as the number of sides goes to infinity.