Decagon Calculator — Side, Apothem, Radius, Perimeter & Area

Professional decagon calculator built for mathematicians and engineers. Enter any single measurement (side, apothem, radius, perimeter, or area) and get every remaining property with deterministic rounding.

Known quantity

What do you know?

Linear unit

Value

Display precision

Highlighted quantity

—

Side length (s) —

Apothem / Inradius (a) —

Circumradius (R) —

Perimeter (P) —

Area (A) —

Angles Interior 144°, Central 36°, Exterior 36°

How to use

Select the quantity you know and enter its numerical value along with the units and desired display precision. The calculator instantly derives every remaining decagon metric, including perimeter and area, with consistent rounding.

Methodology

This tool assumes a regular decagon (n=10) and uses exact trigonometric identities to derive side length, apothem, radius, perimeter, and area. Rounding occurs only when formatting the output, leaving the underlying calculations precise.

Interior angle = 144°, central angle = exterior angle = 36°.
Area can also be expressed as ½·P·a = 5·s·a = 5·R²·sin(π/5).
All units remain consistent across outputs (linear units for lengths, squared for area).

Formulas

Derived relationships for a regular decagon (n=10):

\[P = n \cdot s = 10s\]

\[a = \frac{s}{2\tan\left(\frac{\pi}{n}\right)} = \frac{s}{2\tan\left(\frac{\pi}{10}\right)}, \quad R = \frac{s}{2\sin\left(\frac{\pi}{n}\right)} = \frac{s}{2\sin\left(\frac{\pi}{10}\right)}\]

\[A = \frac{n}{4} s^2 \cot\left(\frac{\pi}{n}\right) = \frac{5}{2} s^2 \cot\left(\frac{\pi}{10}\right) = \frac{1}{2} P a\]

\[A = 5 s a = 5 R^2 \sin\left(\frac{\pi}{5}\right)\]

Citations

NIST Digital Library of Mathematical Functions — https://dlmf.nist.gov/4

Wolfram MathWorld — Regular Polygon — https://mathworld.wolfram.com/RegularPolygon.html

CalcDomain resources — https://calcdomain.com/regular-polygon-calculator

Changelog

0.1.0-draft — 2026-01-19 · Initial audit draft and layout redesign.

✓ Verified by Ugo Candido Last Updated: 2026-01-19 Version 0.1.0-draft

Version 1.5.0

Full original guide (expanded)

Why this calculator

Compute every decagon quantity from a single known value.
Unit-aware inputs that respect squared units for area.
Accessible layout with clear formulas, glossary, and worked example references.

Related tools

Audit notes

Version 0.1.0-draft · Last updated 2026-01-19. Derived from published audit material outlining formulas, glossary entries, and the decision to surface interior and central angles alongside lengths.