Regular Polygon Area Calculator
Calculate the area, perimeter, and interior angles of any regular polygon — from equilateral triangles to dodecagons — given just the side length. Perfect for tiling, architecture, and geometry coursework.
Last updated: September 2026
Formula below · 2 sources (mathworld.wolfram.com, Wikipedia) · Updated Sep 2026
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About this calculator
A regular polygon has all sides equal and all interior angles equal. Its area is given by: Area = (n × s²) / (4 × tan(π / n)), where n is the number of sides and s is the side length. This formula comes from dividing the polygon into n congruent isosceles triangles meeting at the center, each with base s and height equal to the apothem a = s / (2 × tan(π / n)). The perimeter is simply P = n × s. Each interior angle measures (n − 2) × 180° / n degrees. As n increases, the polygon approaches a circle with circumference P and area πr².
How to use
Suppose you want the area of a regular hexagon (n = 6) with side length s = 5 units. Apply the formula: Area = (6 × 5²) / (4 × tan(π / 6)) = (6 × 25) / (4 × tan(30°)) = 150 / (4 × 0.57735) = 150 / 2.3094 ≈ 64.95 square units. The perimeter is P = 6 × 5 = 30 units. Choose Hexagon (6 sides) and enter 5 as the Side Length; the calculator shows Area = 64.9519 square units; perimeter = 30 units; interior angle = 120°.
Frequently asked questions
How do you calculate the area of a regular polygon from the side length?
Use Area = (n × s²) / (4 × tan(π / n)), where n is the number of sides and s is the side length. For example, a regular pentagon (n = 5) with side 4 gives Area = (5 × 16) / (4 × tan(36°)) ≈ 80 / 2.906 ≈ 27.53 square units. The formula works for any n ≥ 3. For very large n the polygon closely approximates a circle, and the formula converges to πr².
What is the apothem of a regular polygon and why does it matter?
The apothem is the perpendicular distance from the polygon's center to the midpoint of any side. It equals a = s / (2 × tan(π / n)). The area can also be expressed as Area = ½ × Perimeter × apothem, which parallels the triangle area formula. The apothem is useful in architecture and tiling because it tells you the inradius — the largest circle that fits inside the polygon.
How does the number of sides affect the area of a regular polygon with a fixed side length?
Increasing the number of sides always increases the total area when the side length is held constant, because each additional side adds another triangular wedge. However, the rate of increase slows as n grows. A square (n=4) with side 1 has area 1, a regular hexagon has area ≈ 2.60, and a regular dodecagon (n=12) has area ≈ 11.20 for the same side length — but the shapes become progressively more circular. This is why circular designs maximize enclosed area for a given perimeter.