How this calculator works
Twelve 30° sectors meet at the centre. Bisecting one creates a right triangle with angle 15°, so the apothem is s(2 + √3)/2. Twelve triangles then give the exact area factor 3(2 + √3). Diagonals span two through six sectors; the six-sector type is a diameter of the circumcircle. Its six copies and the twelve copies of each shorter type account for all 54 diagonals. The table gives unit-independent length ratios, while individual readouts allow length conversion.
Formula
For n = 12: P = ns; A = ns²/(4 tan(π/n)); r = s/(2 tan(π/n)); R = s/(2 sin(π/n)); diagonal spanning k edges = s sin(kπ/n)/sin(π/n).
Worked example
A unit-side dodecagon has perimeter 12 and area 3(2 + √3), about 11.19615 square units. Its interior angle is 150° and its six opposite-vertex diagonals span six edges.
Advertisement
Frequently asked questions
Does this cover an irregular dodecagon?
No. These formulas assume twelve equal edges and twelve 150° interior angles. An irregular twelve-sided outline needs more measurements.
Which diagonal should I measure?
A chord across six side steps joins opposite vertices and equals 2R. Shorter chords span two, three, four or five steps; select the correct type to recover the side.
Can I start from the apothem?
Yes. For a dodecagon, s = 2r tan 15° = 2r(2 - √3). The apothem reaches a side midpoint, not a vertex.