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Polygon Apothem Calculator

Find the apothem of a regular polygon.

Result

8.6603 units
259.81 units²

How it is calculated

Enter the number of sides n (3 or more) and the side length s (greater than 0, in any length unit). The apothem is a = s / (2 × tan(π / n)), with the angle π/n in radians. The area follows from A = ½ × (n × s) × a, where n × s is the perimeter.

The tool rounds a non-integer side count to the nearest whole number, so 5.6 is treated as 6. It returns no result for fewer than 3 sides or for a side length of zero or less. Results are labelled “units” and units²; they are in whatever unit you used for s, and the area is in that unit squared.

Worked example

A regular hexagon has n = 6 and s = 10. First, π/6 gives tan(30°) = 0.57735. Then the apothem is a = 10 / (2 × 0.57735) = 8.6603 units. The perimeter is 6 × 10 = 60, so the area is ½ × 60 × 8.6603 = 259.81 units². The result is an apothem of 8.6603 units and an area of 259.81 units².

Try this example →

Common mistakes

  • Typing the radius (centre to corner) instead of the side length. For a hexagon the two happen to be equal, but for other polygons they are not, and the apothem would come out wrong.
  • Confusing the apothem with the circumradius. The apothem reaches the midpoint of a side and is always shorter than the corner distance.
  • Using degrees in a hand check: the formula needs π/n in radians, or 180°/n if your calculator is in degree mode.
  • Entering a fractional number of sides. The tool rounds it, so 5.6 silently becomes a hexagon.
  • Reading the area as plain length units. It is in squared units, so a side in centimetres gives an area in cm².

About this calculator

The apothem of a regular polygon is the perpendicular distance from its center to the midpoint of any side, calculated as a = s / (2·tan(π/n)), where s is the side length and n is the number of sides. Once known, the apothem gives the polygon's area directly via A = ½ × perimeter × apothem — the same logic used to derive a circle's area as a limiting case when n grows very large.

Architects and structural engineers use the apothem when designing polygonal structures like gazebos, pavilions, or geodesic elements, machinists and fabricators use it for CNC and laser-cutting layouts of hexagonal or octagonal parts (including standard hex bolt heads and nuts), and game and graphics developers use it when rendering or generating regular polygons procedurally. Geometry students also use it as a standard tool for computing polygon area without needing to break the shape into triangles by hand.

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