Polygon Calculator - Area, Perimeter, Angles & Diagonals

Regular polygon formulas

All formulas assume a regular polygon: all sides equal length, all angles equal.

• Perimeter: P = n·s
• Area: A = (n·s²) / (4·tan(π/n))
• Interior angle: α = (n−2)·180° / n
• Exterior angle: β = 360° / n
• Diagonals: D = n·(n−3) / 2
• Inradius (apothem): r = s / (2·tan(π/n))
• Circumradius: R = s / (2·sin(π/n))

Common polygons

• Triangle (3 sides), interior angle: 60°
• Square (4 sides), interior angle: 90°
• Pentagon (5 sides), interior angle: 108°
• Hexagon (6 sides), interior angle: 120°
• Octagon (8 sides), interior angle: 135°

Inradius vs circumradius

The inradius (apothem) is the distance from the center to the midpoint of a side : the radius of the inscribed circle. The circumradius is the distance from the center to a vertex: the radius of the circumscribed circle.

Irregular polygon area: the Shoelace formula

When you know the vertex coordinates of any polygon (regular or irregular), the Shoelace formula (Gauss’s area formula) gives the exact area:

A = ½ |∑(xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|

List the vertices in order (either clockwise or counterclockwise), multiply each x by the next y and subtract the reverse product, sum all terms, and take half the absolute value. This complements the regular polygon formulas above and handles any shape whose vertices are known.

Polygon tilings

Only three regular polygons tile the plane by themselves: the equilateral triangle, the square, and the regular hexagon. The reason: the interior angle of the tiling polygon must divide evenly into 360°. Triangles (60°), squares (90°), and hexagons (120°) satisfy this; pentagons (108°) and others do not.

Named polygons reference