How this calculator works
The centre divides a regular decagon into 10 equal isosceles triangles. Their half-angle gives the side-to-radius relationship. Each diagonal is a chord spanning a whole number of edges. The chord table reports length relative to the side, so it applies in any unit. Opposite-vertex chords occur half as often because counting from each vertex counts the same chord twice.
Formula
For n = 10: 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 decagon has perimeter 10 and area (5/2)√(5 + 2√5), about 7.69421 square units. Its interior angle is 144° and its longest diagonal spans five edges.
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Frequently asked questions
Does this cover an irregular decagon?
No. Equal sides and equal angles are required; side count alone does not determine area.
Which diagonal should I measure?
Count the edges between its endpoints along the shorter boundary route, then choose that separation. Different separations give different chord lengths.
Can I start from the apothem?
Yes. It is the perpendicular centre-to-side distance. It differs from the larger centre-to-vertex circumradius.