Dodecagon Calculator

Dedicated tool for a regular 12‑sided polygon: enter side / circumradius / inradius / area / perimeter – instantly get all properties, with exact formulas and interactive diagram.

Key formulas for n=12: interior angle = 150° , central angle = 30° ; side s = 2R sin(π/12) ; inradius r = R cos(π/12) ; area A = 3 s² cot(π/12) = 6 R² sin(π/6) = 3 R² (since sin(π/6)=1/2, A = 3R²). Actually A = ½·12·R²·sin(30°) = 6R²·½ = 3R².

Dodecagon, cannot be changed
Positive number
Dodecagon properties (live)
Sides (n)
12
Side (s)
0.518
Perimeter (P)
6.212
Area (A)
2.777
Circumradius (R)
1.000
Inradius (r)
0.966
Interior angle
150.0°
Central angle
30.0°
dodecagon
circumcircle (R)
incircle (r)
vertices

Regular Dodecagon – In‑depth Analysis

A regular dodecagon is a polygon with 12 equal sides and 12 equal angles. The name comes from Greek dōdeka (twelve) and gōnia (angle). With its 12-fold symmetry, the dodecagon appears frequently in architecture, coinage, and sacred geometry. It is also constructible with compass and straightedge because 12 = 2²·3, where 3 is a Fermat prime.

? Basic Geometric Properties

  • Interior angle: (12−2)×180° / 12 = 150°. Each interior corner is 30° away from a straight line.
  • Exterior angle: 360° / 12 = 30°.
  • Central angle: also 30° – the angle subtended by each side at the center.
  • Number of diagonals: n(n−3)/2 = 12×9/2 = 54.
  • Symmetry group: Dihedral group D12, with 12 rotational symmetries (every 30°) and 12 reflection axes.
  • Schläfli symbol: {12} for the convex form, and {12/5}, {12/7} for star polygons (since 5 and 7 are coprime to 12).
Exact algebraic expressions (n=12):
sin(π/12) = sin 15° = (√6 – √2)/4 ≈ 0.258819
cos(π/12) = cos 15° = (√6 + √2)/4 ≈ 0.9659258
For unit circumradius (R=1): side s = 2 sin(π/12) = (√6 – √2)/2 ≈ 0.517638,
inradius r = cos(π/12) = (√6 + √2)/4 ≈ 0.9659258,
area A = ½·12·R²·sin(30°) = 6·½ = 3 (since sin30°=½) → A = 3 (exactly!) for R=1.

? Formula Table (n=12)

GivenSide sCircumradius RInradius rArea APerimeter P
side s—s / (2 sin(π/12)) = s·(√6+√2)/2s cot(π/12)/2 = s·(2+√3)/23 s² cot(π/12) = 3 s² (2+√3)12 s
circumradius R2R sin(π/12) = R·(√6–√2)/2—R cos(π/12) = R·(√6+√2)/43 R² (exact)24 R sin(π/12) = 12R·(√6–√2)/2
inradius r2r tan(π/12) = 2r·(2–√3)r / cos(π/12) = 4r/(√6+√2)—12 r² tan(π/12) = 12r²(2–√3)24 r tan(π/12) = 24r(2–√3)
area A√(A tan(π/12) / 3) = √(A(2–√3)/3)√(A/3)√(A / (12 tan(π/12))) = √(A/(12(2–√3)))—12 s

Note: π/12 = 15°. Values like (2+√3) and (2–√3) appear frequently in dodecagon geometry.

? Compass‑and‑straightedge Construction

Since 12 = 2²·3, and 3 is a Fermat prime (2²ⁿ+1 with n=0), a regular dodecagon can be constructed. One classic method: construct a regular hexagon (by inscribing a circle with radius equal to the side), then bisect each central angle (60°/2 = 30°) to obtain the 12 vertices. Alternatively, one can construct a square and an equilateral triangle on the same circle and combine them. The construction was known to Euclid and is detailed in Book IV of the Elements.

? Radical Forms and Special Values

The exact values for sin15° and cos15° are derived from the half‑angle formulas:
cos30° = √3/2, then cos15° = √((1+cos30°)/2) = √((1+√3/2)/2) = (√6+√2)/4.
Similarly sin15° = √((1−cos30°)/2) = (√6−√2)/4.
These expressions are used in the calculator to ensure high precision.

? Appearances in Architecture, Coins and Nature

  • Coins: The British pre‑decimal threepence and the Australian 50‑cent coin are regular dodecagons, chosen for easy recognition by touch.
  • Architecture: The Dome of the Rock in Jerusalem has a dodecagonal exterior; many Gothic rose windows are based on 12‑fold symmetry.
  • Sacred geometry: The dodecagon often represents completeness (12 months, 12 zodiac signs).
  • Games: Some board games (e.g., backgammon boards sometimes incorporate dodecagonal patterns).

? Numerical Benchmarks (R=1, s=1, A=1)

ReferenceSide sCircumradius RInradius rArea A
R=1 (unit circumradius)0.5176410.965933.00000
s=1 (unit side)11.931851.8660311.19615
A=1 (unit area)0.298860.577350.557681

? Approximation of π and the Circle

A regular dodecagon inscribed in a unit circle has perimeter P = 24 sin(π/12) ≈ 24·0.258819 = 6.21266, while the circle's circumference is 2π ≈ 6.28319. The relative error is about 1.12%. Its area is exactly 3 (for R=1), compared to π ≈ 3.14159 – an error of 4.5%. Archimedes used 96‑gons to obtain tighter bounds, but the dodecagon already gives a decent approximation.

? Star Polygons {12/5} and {12/7}

Connecting every 5th vertex of a dodecagon yields the star {12/5} (a twelve‑pointed star with 5 steps), and every 7th gives {12/7} (which is the same as {12/5} traced in the opposite direction). These stars appear in Islamic art and on some flags (e.g., the flag of Malaysia features a 14‑point star, but 12‑point stars are common in heraldry).

❓ Frequently Asked Questions

It has 12 axes of symmetry: 6 through opposite vertices and 6 through midpoints of opposite sides.

Exactly 3. Because A = ½·12·1²·sin(30°) = 6·½ = 3. No approximation needed.

No, because its interior angle 150° does not divide 360° evenly (360/150 = 2.4). However, it can be combined with equilateral triangles and squares to form semi‑regular tilings, e.g., the truncated hexagonal tiling uses dodecagons, hexagons and squares.

From the formula r = (s/2) cot(π/12) = (s/2)(2+√3) ≈ 1.866 s. So the inradius is almost twice the side length.

The interactive calculator on the left lets you explore these properties with your own inputs.