How to use this calculator
- Identify the measurements labelled Apothem r. Check whether the figure satisfies the assumptions described below.
- Enter positive measurements in a common unit. Use the degree symbol convention for angles. Ratios such as eccentricity have no unit.
- Select Calculate. Inspect the result labels: lengths, areas and volumes have different dimensions. Copy the output or print the page if you need a record.
The formula
a = 2r tan(π/n)
Apothem r = 5
Worked example
For Apothem r = 5 cm, applying the displayed relationship gives side ≈ 7.2654253 cm. The remaining related measurements appear in the result panel. The example uses ideal dimensions and rounds only for display.
Model and limitations
Each interior angle is 108 degrees and each central angle is 72 degrees. Joining the centre to the vertices creates five congruent isosceles triangles. Their shared perpendicular height is the apothem.
The apothem reaches a side midpoint; the circumradius reaches a vertex. They are different lengths. Inputs are limited to 10⁹ for lengths, 10¹⁸ for areas and 10²⁷ for volumes. Extremely thin or nearly degenerate figures may lose numerical accuracy; avoid treating displayed rounding as a measurement tolerance.