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Regular Dodecagon Dimensions & Five Diagonal Lengths

Relate a regular twelve-sided shape’s side, perimeter, area, radii and five distinct diagonal spans.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A regular 12-sided outline with side 2 cm has perimeter 24 cm. Its area follows by dividing it into 12 congruent triangles from the center, each with base 2 cm and height equal to the apothem.

P = na; r = a / [2 tan(π/n)]; R = a / [2 sin(π/n)]; A = Pr/2; α = (n−2)π/n; β = 2π/n

Worked example

Enter these known values and leave the other values blank.

Equal side length a
2 cm
Perimeter around the outline
24 cm
Enclosed area
44.785 cm²
Outer circle radius R
3.864 cm
Apothem / inner circle radius r
3.732 cm
Interior angle α
150 deg
Central / exterior angle β
30 deg
Diagonal spanning 2 edges
3.864 cm
Diagonal spanning 3 edges
5.464 cm
Diagonal spanning 4 edges
6.692 cm
Diagonal spanning 5 edges
7.464 cm
Diagonal spanning 6 edges
7.727 cm

Assumptions and limitations

  • The outline must be a convex regular polygon: every side has equal length and every interior angle is equal. An irregular outline needs additional measurements.
  • Use a shared length unit for linear measurements. Area uses the square of a length unit, so check its separate unit selector.
  • The apothem is the perpendicular center-to-edge distance; the circumradius reaches a vertex. They are different quantities.
  • The central angle β equals the exterior turning angle, while the interior angle is α. Their sum is 180°.
  • Reverse calculations require enough independent measurements. Keep the side count locked when the intended shape must remain fixed.
  • A diagonal spanning k edges joins vertices k steps apart. These fixed-shape diagonals appear only at the named side count; their general chord length is 2R sin(kπ/n).

Common questions

Why does area equal half the perimeter times the apothem?

Segments from the center to each vertex partition the polygon into congruent triangles. Adding their base times height divided by two gives the perimeter times the apothem divided by two.

What changes when all lengths double?

Perimeter, radii and diagonals double, while area becomes four times as large. Interior and central angles stay the same.

References

Bookify surfaces invalid derived side counts beside the visible side-length field, including when the side-count control is hidden.

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