Decagon Area Calculator — Regular 10-Sided Polygon
Enter the side length of a regular decagon to get its area, perimeter, circumradius, inradius and longest diagonal — using the exact geometric formulas for a regular 10-sided polygon.
Area = (5/2) × s² ÷ tan(π/10) ≈ 7.6942 × s²
- 1
Square the side
s² = 5² = 25 - 2
Area coefficient
(5/2) ÷ tan(18°) = 7.6942This constant is the same for every regular decagon regardless of scale. - 3
Area
7.6942 × 25 = 192.3552
How does this calculator work?
Area of a regular decagon = (5/2) × s² / tan(18°) ≈ 7.694 s². Perimeter = 10s. Circumradius R ≈ 1.618s (= golden ratio × s exactly); Inradius r ≈ 1.539s. Interior angles = 144° each. The longest diagonal = 2R ≈ 3.236s. Applies only to a regular decagon (all sides and angles equal).
Formula
How this is calculated
A regular decagon is a ten-sided polygon where all sides are equal in length and all interior angles are equal at 144°. The shape appears in architectural ornaments, coin faces, and Islamic geometric patterns.
The area formula A = (n/4) × s² × cot(π/n) applies to any regular n-gon by dividing it into n congruent isosceles triangles from the centre. For the decagon (n = 10), this simplifies to A = (5/2) × s² / tan(18°) ≈ 7.6942 s². The circumradius R — distance from centre to vertex — equals s / (2 sin 18°) ≈ 1.618 s, which is exactly the golden ratio φ times s. The inradius r (apothem, distance from centre to midpoint of a side) equals s / (2 tan 18°) ≈ 1.539 s.
The decagon has 35 diagonals in total, grouped into 4 distinct lengths. The longest diagonal connects two directly opposite vertices and equals 2R (the diameter of the circumscribed circle). To use the circumradius or inradius as input instead of the side, rearrange: s = 2R sin(π/10) or s = 2r tan(π/10).
Frequently asked questions
A = (5/2) × s² ÷ tan(18°) ≈ 7.6942 × s², where s is the side length. This comes from the general regular n-gon formula A = (n/4) × s² × cot(π/n) with n = 10.
The circumradius R is the distance from the centre to a vertex (the circumscribed circle). The inradius r (apothem) is the perpendicular distance from the centre to a side midpoint (the inscribed circle). For a decagon: R = s/(2 sin 18°) ≈ 1.618s; r = s/(2 tan 18°) ≈ 1.539s. R is always larger than r.
For a regular decagon, R = s/(2 sin 18°). Since sin 18° = (√5−1)/4 = 1/(2φ), we get R = s × φ exactly, where φ = (1+√5)/2 ≈ 1.618 is the golden ratio. The decagon has deep connections to the golden ratio — the same constant that appears in pentagons, Fibonacci sequences, and many natural growth patterns.
Also known as
TG we-Calculate Editorial Team. (2026). Decagon Area Calculator — Regular 10-Sided Polygon [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/decagon-area-calculator
TG we-Calculate Editorial Team. "Decagon Area Calculator — Regular 10-Sided Polygon." TG we-Calculate. 2026. https://we-calculate.com/calculator/decagon-area-calculator.
TG we-Calculate Editorial Team, "Decagon Area Calculator — Regular 10-Sided Polygon," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/decagon-area-calculator
@misc{wecalculate_decagon_area_calculator, title = {Decagon Area Calculator — Regular 10-Sided Polygon}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/decagon-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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