Regular polygon apothem calculator from circumradius

Enter the number of sides and the circumradius, which is the distance from the center to a vertex. The calculator applies the cosine relationship directly and returns the apothem as the inradius because those two measurements are identical for every regular polygon. Results can be rounded from zero to twelve decimal places, making the tool useful for quick geometry checks, construction layouts, diagrams, classroom exercises, and repeatable software calculations.

Understand the apothem and circumradius relationship

A regular polygon has equal sides, equal interior angles, and a well-defined center. Draw a line from the center to a vertex and another line from the center perpendicular to the neighboring side. Together with half of that side, these lines form a right triangle. The center-to-vertex line is the circumradius and acts as the hypotenuse. The perpendicular center-to-side line is the apothem. At the center, the angle inside this half-triangle is π divided by the number of sides. Cosine relates the adjacent side to the hypotenuse, so the calculation is apothem = circumradius × cos(π / number_of_sides). The apothem is also the polygon's inradius: it is the radius of the largest circle centered within the polygon that touches every side. This calculator returns both names so results remain clear whether your source material discusses polygon geometry, an inscribed circle, fabrication clearances, or architectural dimensions. No side length, perimeter, or area is required for this particular relationship.

Enter valid measurements and interpret the result

Set number_of_sides to a whole number from 3 through 1,000,000. Three describes an equilateral triangle, four a square, six a regular hexagon, and larger values increasingly resemble a circle. Set circumradius to a finite positive number no greater than 1,000,000,000,000,000. The calculation does not assign a physical unit, so the output uses the same unit as the circumradius: meters in produces meters out, inches in produces inches out, and pixels in produces pixels out. The optional decimals setting controls display rounding from zero through twelve places and defaults to six. Both apothem and inradius contain the same rounded result. The response also repeats number_of_sides and circumradius, which helps automated workflows associate an answer with its inputs. Rounding is applied only after evaluating the cosine formula, preserving the best available precision during calculation. Keep more decimal places for downstream calculations and round more aggressively only for presentation, labeling, or tolerances that genuinely need fewer digits.

Check behavior across different regular polygons

The formula behaves predictably across the supported range. For a square, cos(π/4) is about 0.7071, so the apothem is roughly seventy-one percent of the circumradius. For a regular hexagon, cos(π/6) is about 0.8660, placing each side farther from the center relative to the vertices. As the number of sides increases, π divided by that count becomes smaller and its cosine approaches one. The apothem therefore approaches the circumradius, matching the intuition that a many-sided regular polygon increasingly resembles its circumscribed circle. A valid result is always positive and never greater than the supplied circumradius. If a workflow produces the opposite relationship, check whether the supplied value was actually a diameter, side length, or apothem rather than a circumradius. The algorithm is deterministic, uses no network access, and depends only on the submitted numbers. Identical inputs produce identical JSON results, which makes the capability suitable for test fixtures, spreadsheets, CAD helpers, educational systems, and server-side validation where reproducibility matters.