Decagon Calculator

Compute side length, circumradius, inradius, area, perimeter, interior angle of a regular decagon. Instant real‑time update with interactive diagram.

Decagon specific: side s = 2R·sin(18°) ≈ 0.618·2R? Actually sin(π/10)=sin18°≈0.3090 → s = 0.618R·? Wait: s = 2R sin(π/10) ≈ 0.618R. Inradius r = R cos(18°) ≈ 0.9511R. Area A = (5/2) s² cot(18°) ≈ 7.694 s². Also golden ratio φ = (1+√5)/2 appears in decagon.

Fixed for decagon (10 sides)
Positive number
Regular decagon: 10 equal sides, interior angle 144°, central angle 36°. Often related to golden ratio (φ = 1.618...).
Decagon properties
Sides (n)
10
Side length (s)
1.000
Perimeter (P)
10.000
Area (A)
7.694
Circumradius (R)
1.618
Inradius (r)
1.539
Interior angle
144.0°
Central angle
36.0°
decagon
circumcircle (R)
incircle (r)
vertices

Understanding the Regular Decagon

A regular decagon is a 10‑sided polygon with all sides equal and all interior angles equal (144°). It is highly symmetric and appears in architecture, art, and nature (e.g., some flowers and starfish).

? Key formulas (n=10, side s, circumradius R, inradius r, area A):

  • Interior angle = (10-2)×180°/10 = 144°
  • Central angle = 360°/10 = 36°
  • Circumradius: R = s / (2 sin18°) = s · φ (since 1/(2 sin18°) = φ ≈ 1.618)
  • Inradius: r = R cos18° = (s/2) cot18° ≈ 1.539 s
  • Area: A = ½ n R² sin36° = ¼ n s² cot18° = n r² tan18°
  • Exact values: sin18° = (√5−1)/4, cos18° = √(10+2√5)/4, cot18° = √(5+2√5)

✨ The Golden Ratio Connection

The regular decagon is deeply linked to the golden ratio φ = (1+√5)/2 ≈ 1.618034. Observe:

  • Circumradius to side: R / s = φ
  • Diagonal to side: The diagonal connecting two vertices with two steps between them equals s·φ. (In a regular decagon, there are four distinct diagonal lengths; the longest equals s·φ.)
  • Construction: To construct a regular decagon, first construct a pentagon (which itself involves φ), then bisect its central angles.

Mathematically, sin18° = (√5−1)/4, so 1/(2 sin18°) = 2/(√5−1) = (√5+1)/2 = φ.

? Numerical Properties (s = 1)

PropertyValueFormula
Side length s1-
Perimeter P1010·s
Circumradius R≈1.6180s·φ
Inradius r≈1.5388½ s·cot18°
Area A≈7.6942¼·10·s²·cot18°
Diagonal (short)≈1.6180s·φ (two-step)
Diagonal (long)≈1.9021s·√(2+2cos36°) ...

?️ Constructibility

A regular decagon is constructible with compass and straightedge because 10 = 2 × 5, and both 2 and 5 are Fermat primes (5 is a Fermat prime). The construction relies on the golden ratio: first construct a regular pentagon, then draw its circumcircle and bisect the arcs to obtain the decagon vertices.

? Real‑world applications

  • Architecture: Decagonal buildings, domes, and decorative tiles (e.g., some historical mosques).
  • Art & Design: Islamic geometric patterns, coins (some currencies have decagonal shape, like the Chinese “10 yuan” coin).
  • Chemistry: Some molecular structures (e.g., decametallic complexes, fullerene fragments).
  • Mathematics: Used in tiling theory and as a base for constructing larger polygons.

Frequently Asked Questions

A 10‑sided polygon where all sides are equal and all interior angles are 144°. It can be inscribed in a circle (circumcircle) and has an inscribed circle (incircle).

The ratio of the circumradius to the side length is exactly the golden ratio φ = (1+√5)/2 ≈ 1.618. Also, the length of the diagonal that skips one vertex equals s·φ.

Approximately 7.6942 square units. The exact expression involves cot(18°) which is √(5+2√5). So A = (5/2) √(5+2√5) ≈ 7.6942.

Yes, because 10 = 2 × 5, and 5 is a Fermat prime. Therefore a regular decagon is constructible. The construction involves the golden ratio and can be derived from a regular pentagon.