Regular Polygon Calculator

A regular polygon is any flat shape with all sides equal and all angles equal, from the triangle and square up through the pentagon, hexagon, octagon and beyond, and this calculator works out every key property of one from just two inputs: how many sides it has and how long each side is. Enter those, and it returns the area, the perimeter, the interior and exterior angles, the apothem and the circumradius, all updating as you type. The beauty of regular polygons is that the same handful of formulas works for every one of them, because each can be divided into identical triangles meeting at the centre. The interior angle is the number of sides minus two, times 180, divided by the number of sides, so it grows from 60 degrees for a triangle towards 180 as the polygon gains sides and starts to resemble a circle. The exterior angle is simply 360 divided by the number of sides. The area can be found as half the perimeter times the apothem, the distance from the centre to the middle of a side, or directly from the side length and a tangent term. The circumradius, from the centre to a corner, is the radius of the circle that passes through every vertex. That makes this tool genuinely useful for geometry and trigonometry students checking work and seeing how the shapes relate, for engineers, architects and designers laying out polygonal forms, for tilers and crafters planning patterns, and for anyone needing the dimensions of a nut, a bolt head, a paving stone or a garden bed. Because the results recalculate instantly, you can increase the number of sides and watch the polygon's area approach that of its circumscribed circle. The formulas and a worked example are explained clearly below.

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