Regular Polygon Calculator
For a regular polygon with any number of equal sides, get its area, perimeter, interior angle and apothem.
How it works
Area = ¼ · n · s² · cot(π ÷ n) · Perimeter = n · s · Interior angle = (n − 2) · 180° ÷ n · Apothem = s ÷ (2 · tan(π ÷ n))
A regular polygon has all sides and all angles equal. Split it from the centre into n identical isosceles triangles, each with base s and height equal to the apothem (the perpendicular distance from centre to a side). Summing their areas gives ¼·n·s²·cot(π ÷ n). The interior angle grows toward 180° as the sides multiply, which is why many-sided polygons look almost circular.
Worked example
A regular hexagon (n = 6) with side s = 4: apothem = 4 ÷ (2·tan 30°) ≈ 3.4641, perimeter = 6 × 4 = 24, and area = ¼ × 6 × 16 × cot 30° = 24 × 1.7320 ≈ 41.5692. Each interior angle is (6 − 2) × 180° ÷ 6 = 120°.
Sources: Regular polygon — Wikipedia
Frequently asked questions
What is the apothem?
The apothem is the perpendicular distance from the centre of the polygon to the midpoint of any side — the radius of the inscribed circle. Area can also be written as ½ × perimeter × apothem, which is why the apothem matters.
How is the interior angle found?
The interior angles of any n-gon sum to (n − 2) × 180°. Dividing by n gives each equal angle in a regular polygon: 60° for a triangle, 90° for a square, 108° for a pentagon, 120° for a hexagon, and so on.
Why must the number of sides be at least 3?
A closed polygon needs a minimum of three sides to enclose any area. With one or two sides there is nothing to bound a region, so the formulas only make sense for n ≥ 3.
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