Dodecagon Calculator

Enter a regular dodecagon's side length, apothem, or circumradius to get its area, perimeter, apothem, and circumradius, using A = 3(2 + √3)s².

Quick Facts

Interior Angle
150°
(n − 2) × 180° / n for n = 12; each interior angle of a regular dodecagon measures 150°.
Area from Side
A = 3(2 + √3)s²
≈ 11.1962 × s² — the exact area formula for a regular 12-gon of side length s.
Area from Circumradius
A = 3R²
A clean special-case identity: three times the square of the circumscribed-circle radius.
Diagonals
54
n(n − 3) / 2 = 12 × 9 / 2 total diagonals connect the 12 vertices.

Your Results

Calculated
Area
-
A = 3(2 + √3) × side², in square units
Perimeter
-
P = 12 × side
Apothem (Inradius)
-
Distance from center to the midpoint of a side
Circumradius
-
Distance from center to each vertex

Ready

Choose a known measurement, enter its value and unit, then press Calculate.

Formula and Method for the Dodecagon Calculator

A dodecagon is a 12-sided polygon. This calculator focuses on the regular dodecagon, where all 12 sides have equal length and all 12 interior angles equal 150°. Because a regular dodecagon is fully determined by a single length measurement, entering any one of its side length, apothem (inradius), or circumradius lets the calculator derive the other two measurements, plus the perimeter and area.

How the calculation works

Enter a known measurement and choose which one it is: side length, apothem, or circumradius. The calculator first converts that value to the side length s. If you entered the apothem a, it uses s = 2a·tan(15°); if you entered the circumradius R, it uses s = 2R·sin(15°), where 15° = 180°/12 is half the central angle subtended by one side. From s, it computes the perimeter as P = 12s, the apothem as a = s / (2·tan(15°)), the circumradius as R = s / (2·sin(15°)), and the area as A = 3(2 + √3)s² ≈ 11.1962s² — equivalently, A = ½ × perimeter × apothem, the same half-base-times-height identity used for any regular polygon. As a special case, area expressed from the circumradius alone simplifies to the clean identity A = 3R².

Common mistakes

  • Confusing apothem and circumradius: the apothem (inradius) touches the midpoint of a side and is always shorter than the circumradius, which reaches a vertex.
  • Selecting the wrong measurement type: if you enter a circumradius value while "Side Length" is still selected, every derived result will be wrong even though the arithmetic itself is correct.
  • Mixing units: keep the entered length in one consistent unit — convert millimeters to centimeters, or inches to feet, before entering the value.

Real-world applications

  • Coin design: the pre-2017 UK three-pence piece and the current UK £1 coin are both minted as regular dodecagons, partly because a 12-sided shape is easy to identify by touch and hard to counterfeit.
  • Architecture and tiling: dodecagonal floor medallions, gazebos, and paving stones rely on the interior-angle and side-length relationships to cut accurate mitred edges.
  • Machining and hardware: dodecagon-head bolts and sockets use the same apothem-versus-circumradius geometry that separates a wrench's flat-to-flat size from its point-to-point size.
  • Fabrication and prototyping: laser-cutting or 3D-printing a 12-sided panel needs the exact side length to hit a target area or a target circumradius footprint.

Frequently Asked Questions

What is a dodecagon?
A dodecagon is a 12-sided, 12-angled polygon. In a regular dodecagon, all 12 sides are equal in length and all 12 interior angles measure exactly 150° each, for a total interior angle sum of 1800°.
What is the formula for the area of a regular dodecagon?
The area of a regular dodecagon with side length s is A = 3(2 + √3)s² ≈ 11.1962 × s². For example, a dodecagon with 5 cm sides has an area of about 11.1962 × 5² ≈ 279.9 cm².
How do I find the area from the apothem or circumradius instead of the side length?
From the apothem a, area is A = 12a²tan(15°) ≈ 3.2154 × a². From the circumradius R, the formula simplifies to A = 3R² exactly. This calculator converts whichever measurement you enter into the side length first, then derives area, perimeter, apothem, and circumradius from that.
What is the interior angle of a dodecagon?
Each interior angle of a regular dodecagon measures 150°, from the polygon formula (n − 2) × 180° / n with n = 12. The exterior angles each measure 30° and sum to 360°, as they do for any convex polygon.

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