Polygon Calculator

A regular polygon has all sides equal in length and all interior angles equal. Examples include equilateral triangles (3 sides), squares (4), regular pentagons (5), hexagons (6), and so on.

The area formula is A = (n × s²) / (4 × tan(π/n)), where n is the number of sides and s is the side length. Equivalently, A = ½ × P × a, where P is the perimeter and a is the apothem.

Each interior angle is θ = (n-2) × 180° / n. For n = 3 (triangle): 60°. For n = 4 (square): 90°. For n = 6 (hexagon): 120°. As n increases, the angle approaches 180°.

A polygon with n sides has n(n-3)/2 diagonals. For example, a pentagon has 5 diagonals, a hexagon has 9, and a decagon has 35.

The apothem is the perpendicular distance from the center to the midpoint of a side: a = s/(2tan(π/n)). It is used in the area formula A = ½ × P × a.

The circumradius is the distance from the center to any vertex: R = s/(2sin(π/n)). The polygon can be inscribed in a circle of this radius.

The inradius equals the apothem: r = s/(2tan(π/n)). It is the radius of the largest circle that fits inside the polygon (the incircle).

Rearrange the area formula to solve for n iteratively, or use n = 4tan(π/n) × A / s². Since n appears in both the formula and the tangent argument, numerical methods are typically needed.

The sum is (n-2) × 180°. For n = 5: 540°. For n = 6: 720°. For n = 10: 1440°. Each additional side adds 180° to the total.

Each exterior angle is 360° / n. It is the supplement of the interior angle. For a regular hexagon: 360° / 6 = 60°. The sum of all exterior angles is always 360°.

A regular polygon can have any integer number of sides n ≥ 3. As n → ∞, the regular polygon approaches a circle. Constructible regular polygons are limited to certain values (Fermat primes).

The perimeter is simply P = n × s, where n is the number of sides and s is the side length. If you know the apothem, s = 2a × tan(π/n), so P = 2na × tan(π/n).

The apothem and circumradius are related by: a = R × cos(π/n). The apothem is always less than the circumradius (they are equal only in the limit as n → ∞).

As n increases, the polygon's area approaches πR² and its perimeter approaches 2πR. This is how Archimedes estimated π — by inscribing and circumscribing polygons around a circle with increasing n.

The ratio A/P = a/2 (half the apothem). For a given circumradius, larger polygons (more sides) have a higher area-to-perimeter ratio, approaching the circle's optimal ratio of R/2.

Gauss proved that regular n-gons are constructible with compass and straightedge when n = 2^k × product of distinct Fermat primes. The first few are: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17...

All interior angles of a convex polygon are less than 180°. A concave polygon has at least one interior angle greater than 180°, creating an inward "dent." Regular polygons are always convex.

A regular polygon can be divided into n congruent isosceles triangles from the center. Each triangle has base s and height a (apothem). Area = n × ½ × s × a = ½ × P × a = (n × s²) / (4tan(π/n)).

For a regular pentagon with side s: a = s/(2tan(36°)) ≈ 0.6882s. The interior angle is 108°, and the apothem is about 68.8% of the side length.

A star polygon {n/k} is formed by connecting every kth vertex of a regular n-gon. Examples include the pentagram {5/2} and hexagram {6/2}. Star polygons have different area and perimeter formulas than convex polygons.

A regular hexagon with side s has area A = (3√3/2) × s² ≈ 2.5981 × s². This simplifies from the general formula since tan(π/6) = 1/√3.

A regular pentagon (5 sides) has interior angles of 108° each. The sum of all interior angles is (5-2) × 180° = 540°. The exterior angle is 72°.

Using the apothem a and side length s: A = ½ × P × a = ½ × (n × s) × a. Alternatively, if you only know the apothem, use s = 2a × tan(π/n) to find the side first.

A regular polygon can be inscribed in a circle (circumcircle) or have a circle inscribed within it (incircle). As the number of sides increases, the polygon approximates the circle more closely, converging to πR² for area and 2πR for perimeter.

The Polygon Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Polygon Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.