Calculate regular n-sided polygon area, perimeter, apothem, circumradius, interior angles, and exterior angles with step-by-step steps.
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A regular polygon is an n-sided polygon that is both equiangular (all angles are equal) and equilateral (all sides have equal length). As the number of sides approaches infinity, the regular polygon approaches a circle.
A = \frac{n s^2}{4\tan\left(\frac{\pi}{n}\right)}, \quad a = \frac{s}{2\tan\left(\frac{\pi}{n}\right)}, \quad \theta_{\text{int}} = \frac{(n-2)\times 180^\circ}{n}Area is calculated using trigonometry by dividing the n-gon into n isosceles triangles. Interior angle uses the polygon angle sum theorem.
Scenario: Calculate regular octagon (n = 8) with side length s = 5 cm
Perimeter: P = 8 × 5 = 40 cm
Interior angle: (8 - 2) × 180° / 8 = 1080° / 8 = 135°
Apothem: a = 5 / (2 × tan(180° / 8)) = 5 / (2 × tan(22.5°)) ≈ 6.0355 cm
Area: A = ½ × 40 × 6.0355 ≈ 120.7107 cm²
A polygon must have at least 3 sides (a triangle).
The inradius (apothem) is the radius of the inscribed circle touching the sides; the circumradius is the radius of the circle passing through the vertices.
The apothem is a line segment from the center of a regular polygon perpendicular to one of its sides.
The sum of interior angles is always (n - 2) × 180 degrees.
Calculate regular pentagon area, perimeter, apothem, diagonals, and interior angles from side length with step-by-step math.
Calculate regular hexagon area, perimeter, apothem, long diagonal, short diagonal, and circumradius with step-by-step steps.
Calculate circle area, circumference, diameter, radius, arc length, and sector area with step-by-step geometric derivations.