Decagon Calculator — Area, Perimeter & Diagonals
Find every dimension of a regular decagon — area, perimeter, circumscribed circle radius, inscribed circle radius (apothem), longest diagonal and more — from the side length, circumradius or apothem.
Known measurement
Area = (10 × s²) / (4 × tan 18°) ≈ 7.6942 × s²
- 1
Square the side
s² = 5² = 25 - 2
Area coefficient
10 ÷ (4 × tan(18°)) = 7.6942The general regular-polygon area factor (10 sides, π/10 = 18°). - 3
面积
7.6942 × 25 = 192.3552
这个计算器是如何工作的？
Regular decagon: Area = (10 s²) / (4 tan 18°) ≈ 7.694 s²; Perimeter = 10 s; Circumradius R ≈ 1.618 s (the golden ratio!); Inradius r ≈ 1.539 s; 35 diagonals; all interior angles = 144°. Enter any one measurement to get all others.
公式
How this is calculated
A regular decagon has 10 sides of equal length s and 10 equal interior angles of 144°. All its properties follow from the side length using trigonometry on the central triangle formed by two radii and one side. That triangle has a central angle of 360° / 10 = 36°, and half that triangle gives a right triangle with one angle of 18° (π/10).
The circumradius R (centre to vertex) is R = s / (2 sin 18°). Because sin 18° = (√5 − 1) / 4, the ratio R / s equals the golden ratio φ ≈ 1.6180 — a remarkable connection to the regular decagon's relationship with the pentagon. The inradius (apothem) r is the distance from the centre to the midpoint of a side: r = s / (2 tan 18°) ≈ 1.5388 s. The area equals the perimeter times the apothem divided by 2: Area = (10s × r) / 2, which simplifies to (10 s²) / (4 tan 18°).
The decagon has n(n − 3) / 2 = 35 diagonals in total. The longest diagonal connects opposite vertices and equals the diameter 2R. You can enter any one of side length, circumradius, or inradius and the calculator derives all remaining quantities in the same unit.
常见问题
Each interior angle is (10 − 2) × 180° / 10 = 144°. The exterior angle is 36°. All 10 interior angles are equal in a regular decagon.
For a regular decagon with side s, the circumradius R = s / (2 sin 18°) = s × φ where φ = (1 + √5) / 2 ≈ 1.618 is the golden ratio. This arises because sin 18° = (√5 − 1) / 4, and the decagon shares this deep geometric relationship with the regular pentagon.
A regular decagon has n(n − 3) / 2 = 10 × 7 / 2 = 35 diagonals. They come in four distinct lengths (not counting sides), with the longest being the diameter 2R connecting opposite vertices.
也称为
TG we-Calculate Editorial Team. (2026). Decagon Calculator — Area, Perimeter & Diagonals [Online calculator]. TG we-Calculate. https://we-calculate.com/zh/calculator/decagon-calculator
TG we-Calculate Editorial Team. "Decagon Calculator — Area, Perimeter & Diagonals." TG we-Calculate. 2026. https://we-calculate.com/zh/calculator/decagon-calculator.
TG we-Calculate Editorial Team, "Decagon Calculator — Area, Perimeter & Diagonals," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/zh/calculator/decagon-calculator
@misc{wecalculate_decagon_calculator, title = {Decagon Calculator — Area, Perimeter & Diagonals}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/zh/calculator/decagon-calculator}}, year = {2026}, note = {TG we-Calculate} }
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