Area of a Regular Polygon
Calculate the area of any regular polygon using A = (n × s² × cot(pi/n)) / 4.
Works for pentagons, hexagons, and beyond.
The Formula
Or equivalently: A = ½ × n × s × a
The area of a regular polygon — a shape where all sides and all angles are equal — can be calculated using a single formula that works for any number of sides. The first form uses only the number of sides (n) and the side length (s). The second form uses the apothem (a), which is the perpendicular distance from the center to the middle of any side.
A regular polygon can be divided into n identical isosceles triangles, each with its base along one side of the polygon and its apex at the center. Each triangle has a base of length s and a height equal to the apothem a. The area of each triangle is ½ × s × a, and since there are n triangles, the total area is ½ × n × s × a. The apothem can be expressed in terms of s and n as a = s / (2 × tan(π/n)), which gives the first form of the formula.
This formula is remarkably versatile. For a square (n = 4), it simplifies to s². For an equilateral triangle (n = 3), it gives (√3/4) × s². For a regular hexagon (n = 6), it produces (3√3/2) × s². As n increases toward infinity, the regular polygon approaches a circle, and the formula approaches πr² where r is the apothem.
The relationship between regular polygons and circles is mathematically elegant. A regular polygon with n sides can be inscribed in a circle of radius R (the circumradius, from center to vertex) or circumscribed around a circle of radius a (the apothem, from center to side midpoint). These two radii are related by a = R × cos(π/n). The area can also be written as A = ½ × n × R² × sin(2π/n), using the circumradius instead.
Regular polygons appear frequently in architecture, tiling, and engineering. Hexagonal patterns are found in honeycombs, floor tiles, and bolt heads. Pentagon and octagon shapes appear in building floor plans and stop signs. Being able to quickly calculate the area of these shapes is useful in construction, design, and material estimation.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the regular polygon (square units) |
| n | Number of sides (dimensionless, must be 3 or more) |
| s | Length of one side (linear units) |
| a | Apothem — perpendicular distance from center to side midpoint (linear units) |
| R | Circumradius — distance from center to a vertex (linear units) |
Example 1
Find the area of a regular hexagon (n = 6) with side length 10 cm.
A = (6 × 10²) / (4 × tan(π/6))
tan(π/6) = tan(30°) = 1/√3 ≈ 0.5774
A = (6 × 100) / (4 × 0.5774) = 600 / 2.3094
A ≈ 259.81 cm²
Example 2
A regular octagon (n = 8) has a side length of 5 inches. Calculate its area.
A = (8 × 5²) / (4 × tan(π/8))
tan(π/8) = tan(22.5°) ≈ 0.4142
A = (8 × 25) / (4 × 0.4142) = 200 / 1.6569
A ≈ 120.71 in²
When to Use It
The regular polygon area formula is used whenever you need to find the area of an equal-sided shape.
- Calculating floor tile area for hexagonal or octagonal tiles
- Estimating material needed for polygon-shaped tabletops, signs, or windows
- Architecture and landscape design involving regular polygon floor plans
- Engineering calculations for bolt heads, nuts, and other polygon-shaped hardware
- Mathematics education and geometry coursework
- Understanding the relationship between polygons and circles as n approaches infinity