Regular Polygon Area Calculator

Regular Polygon Area Calculator – Any Number of Sides

A regular polygon is a flat, closed shape with any number of sides where every side has the same length and every interior angle is equal. The simplest is the equilateral triangle (3 sides); beyond that come the square (4), pentagon (5), hexagon (6), and so on up to any arbitrarily large n-gon. This calculator computes the enclosed area of any regular polygon from whichever single measurement you have — the side length, the circumscribed circle radius, or the inscribed circle radius — and simultaneously derives all other geometric properties of the shape.

Area Formulas for a Regular Polygon

Three equivalent formulas exist for the area of a regular n-gon, depending on which measurement is known:

• From side length (s): A = (n × s²) / (4 × tan(π/n))
• From circumradius (R): A = (n × R² × sin(2π/n)) / 2
• From apothem (a): A = n × a² × tan(π/n)

All three formulas produce the same result for any given polygon. The calculator converts your input to side length internally and applies the first formula, then derives the remaining properties.

Circumradius vs Apothem

Understanding the two radii of a regular polygon is important for selecting the right input mode:

• Circumradius (R) — the distance from the polygon's center to any vertex (corner). It is the radius of the circumscribed circle — the smallest circle that passes through every corner. Formula: R = s / (2 × sin(π/n)).
• Apothem (a), also called the inradius — the perpendicular distance from the center to the midpoint of any side. It is the radius of the inscribed circle — the largest circle that fits inside the polygon touching every side. Formula: a = s / (2 × tan(π/n)).

The two radii are related by a = R × cos(π/n), so the apothem is always smaller than the circumradius (except in the limiting case of a circle, where they converge).

Derived Properties

From the side length of a regular n-gon, all other properties follow directly:

• Perimeter: P = n × s
• Apothem: a = s / (2 × tan(π/n))
• Circumradius: R = s / (2 × sin(π/n))
• Interior angle: θ = ((n − 2) × 180) / n degrees — for a triangle 60°, for a square 90°, for a hexagon 120°, approaching 180° as n → ∞
• Exterior angle: φ = 360 / n degrees — always sums to 360° regardless of the number of sides
• Sum of interior angles: (n − 2) × 180° — for a triangle 180°, for a square 360°, for a hexagon 720°
• Number of diagonals: n(n − 3) / 2 — a triangle has 0, a quadrilateral has 2, a pentagon has 5, a hexagon has 9

Using the Preset Shape Buttons

The quick-preset buttons at the top of the calculator let you jump directly to any common polygon — triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, or decagon — without typing the side count manually. For polygons with more than 10 sides, simply type the number of sides directly into the "Number of sides" field. The polygon preview SVG updates in real time to show the current shape.

Unit Flexibility

Select any length unit (mm, cm, m, km, in, ft, yd) for your input measurement and independently choose any area unit (mm², cm², m², km², in², ft², yd²) for the result. The calculator converts all lengths to meters internally, computes the area in m², and then converts to your chosen output unit. A conversion table shows the area expressed in every supported unit simultaneously.

Real-World Applications

Regular polygon area calculations appear in many practical contexts. Architects and tile designers need exact areas for hexagonal and octagonal floor tiles and paving slabs. Structural engineers calculate cross-sectional areas for polygonal column profiles and bolt holes. Game developers building hex-grid or octagon-grid maps use polygon area formulas to determine cell coverage and scale game worlds. Manufacturers of nuts, bolts, and fasteners rely on regular hexagon geometry for wrench-size standards. Educators and students use the general formula to build intuition about how polygon shapes evolve as the number of sides increases toward a circle.

Relationship to a Circle

As the number of sides n increases, a regular polygon approaches a circle. The area formula A = (n × s²) / (4 × tan(π/n)) converges to the circle area formula A = π × r² as n → ∞, because the apothem approaches the circle's radius and the perimeter approaches the circumference. A 100-sided regular polygon (hectagon) already has an area within 0.01% of its circumscribed circle.

Accuracy

All computations use IEEE 754 double-precision floating-point arithmetic, giving approximately 15–16 significant digits of accuracy. Display precision is adjustable from 0 to 10 decimal places. For very large polygons (n greater than a few thousand), floating-point rounding in the trigonometric functions may introduce tiny errors, but these are negligible for any practical application.