Polygon Calculator

A regular polygon is a closed two-dimensional shape whose sides all have the same length and whose interior angles all have the same measure. Familiar examples include an equilateral triangle, a square, a regular pentagon, and a regular hexagon. The number of sides determines the overall shape, while the side length controls its scale. This calculator combines those two measurements to find the polygon's perimeter, area, apothem, interior angle, and exterior angle. Perimeter is the simplest property: multiply the number of sides n by the side length s. Area can be found by dividing the polygon into n congruent isosceles triangles that meet at the center. The area formula is n times s squared divided by four times the tangent of pi divided by n. This formula works for every regular polygon with at least three sides. It gives square units, so a side measured in centimeters produces an area in square centimeters. The apothem is the perpendicular distance from the center to the midpoint of any side. It is also the height of each central triangle and equals s divided by two times the tangent of pi divided by n. Area can therefore be checked with the equivalent formula one half times perimeter times apothem. The circumradius, by contrast, reaches from the center to a vertex and is longer than the apothem except as the side count becomes very large. Every regular polygon has equal interior angles. Their total is (n - 2) times 180 degrees, and dividing by n gives each interior angle. Each exterior angle is 360 degrees divided by n. An interior angle and its adjacent exterior angle always add to 180 degrees. For a regular hexagon, for example, each interior angle is 120 degrees and each exterior angle is 60 degrees. Use positive lengths in any consistent unit. The perimeter and apothem retain that unit, while area uses its square. The side count must be a whole number because a polygon cannot have a fractional side. As n increases while side length remains fixed, the polygon grows and increasingly resembles a circle around its center. Regular polygon calculations are useful in geometry homework, tiling, architecture, machining, graphic design, and construction layout. The calculator keeps full floating-point precision internally and rounds only the displayed values, making it convenient for both quick estimates and formula checks.