Dodecagon Calculator

Calculate all properties of a regular dodecagon (12-sided polygon) from its side length.

Enter Side Length

Result

Area
279.9038
square units

Step-by-Step Solution

A = 3s^2(2 + sqrt(3))

What is a Dodecagon?

A dodecagon is a polygon with 12 sides and 12 vertices. A regular dodecagon has all sides equal in length and all interior angles equal to 150 degrees. The dodecagon is one of the most symmetric polygons and appears frequently in architecture, coin design, and tiling patterns.

Dodecagon Formulas

Area

The area of a regular dodecagon with side length s.

A = 3s2(2 + sqrt(3))

Perimeter

The total length of all 12 sides.

P = 12s

Apothem

The distance from center to the midpoint of a side.

a = (s/2) x cot(pi/12)

Circumradius

The distance from center to a vertex.

R = s / (2 x sin(pi/12))

Interior Angle

Each interior angle of a regular dodecagon.

angle = (12-2) x 180/12 = 150 deg

Number of Diagonals

Lines connecting non-adjacent vertices.

d = 12(12-3)/2 = 54

Properties of a Regular Dodecagon

A regular dodecagon has 12 equal sides, 12 equal interior angles of 150 degrees each, and 54 diagonals. The sum of all interior angles is 1800 degrees. It has 12 lines of symmetry and rotational symmetry of order 12.

Dodecagons in Real Life

  • Many coins around the world use dodecagonal shapes (e.g., the British one-pound coin).
  • Dodecagonal floor tiles are used in decorative patterns and architecture.
  • Some clock faces are designed with a dodecagonal outline to match the 12 hours.
  • In engineering, dodecagonal cross-sections appear in certain structural designs.
  • The dodecagon is related to the dodecahedron, a 3D solid with 12 pentagonal faces.

Relationship to Other Polygons

A regular dodecagon can be constructed by combining regular triangles and squares. It can also be seen as a truncated hexagon. As the number of sides increases, regular polygons approach the shape of a circle, and the dodecagon with its 12 sides is already quite close to circular.

How to Construct a Regular Dodecagon

  • Draw a circle with the desired circumradius.
  • Divide the circle into 12 equal parts (every 30 degrees).
  • Connect consecutive division points with straight lines.
  • Each of the 12 resulting segments will be equal in length.