Regular Polygon Calculator
Calculate the area, perimeter, interior angles, and diagonal of any regular polygon.
→ Regular Hexagon
What Is the Regular Polygon Calculator?
The Regular Polygon Calculator computes the area, perimeter, interior angle, exterior angle, apothem (inradius), and circumradius of any regular polygon (equilateral triangle, square, pentagon, hexagon, and beyond). Enter the number of sides and either side length, apothem, or circumradius.
Formula
How to Use
Enter the number of sides n (minimum 3). Enter one known measurement: side length, apothem (radius of inscribed circle), or circumradius (radius of circumscribed circle). The calculator derives all other measurements and shows the polygon's geometric properties.
Example Calculation
Regular hexagon (n=6), side s=5 cm: Area = (6×25)/(4×tan(30°)) = 150/(4×0.5774) = 64.95 cm². Perimeter = 30 cm. Interior angle = (6−2)×180/6 = 120°. Apothem = 5/(2×tan(30°)) ≈ 4.33 cm.
Understanding Regular Polygon
A regular polygon has all sides equal in length and all interior angles equal in measure. The simplest is the equilateral triangle (n=3), followed by the square (n=4), regular pentagon (n=5), hexagon (n=6), and so on. As n increases, the polygon approximates a circle more closely — the inscribed and circumscribed circles converge to the same radius.
The area formula A = ns²/(4tan(π/n)) can be understood geometrically: dividing the polygon into n isoceles triangles from the center, each has base s and height equal to the apothem. Total area = n × (½ × s × apothem) = n × (½ × s × s/(2tan(π/n))) = ns²/(4tan(π/n)).
Regular polygons appear throughout science, engineering, and nature. Hexagons are used in honeycomb structures (maximizing area for a given perimeter). Regular pentagons appear in virus capsid geometry. Octagons are used in stop sign design. In architecture, polygonal floor plans (octagons, hexagons) create efficient, aesthetically appealing spaces. Understanding polygon geometry is essential for tiling, computer graphics, and structural design.
Frequently Asked Questions
What is an apothem?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side — it is the radius of the inscribed circle. For a regular n-gon with side s: apothem = s/(2×tan(π/n)).
What is the interior angle of a regular polygon?
Interior angle = (n−2)×180°/n. For a triangle: 60°, square: 90°, pentagon: 108°, hexagon: 120°, octagon: 135°. As n→∞, angles approach 180° (becoming a circle).
What is the circumradius?
The circumradius R is the distance from the center to any vertex — the radius of the circumscribed circle. R = s/(2×sin(π/n)).
What regular polygons tessellate the plane?
Only three regular polygons tile the plane without gaps: equilateral triangles (6 around each vertex), squares (4 around each vertex), and regular hexagons (3 around each vertex).
Is this calculator free?
Yes, completely free with no registration needed.