How this instrument works
A regular decagon has ten equal sides and ten equal 144° interior angles, and the area formula falls straight out of that symmetry. Draw a line from the centre to every vertex and the shape splits into ten identical isosceles triangles, each with a base of s and two legs meeting at the centre. A full turn is 360°, so each triangle's angle at the centre is exactly 36°; bisecting that angle to solve for the triangle's height, the apothem, is where the formula's tan 18° comes from.
That 18° also ties the decagon to the golden ratio in a way few people notice. Because sin 18° equals (√5 − 1) ⁄ 4, a regular decagon's circumradius — centre to any vertex — works out to exactly φ times its side length, where φ ≈ 1.618034 is the same constant behind the golden rectangle and the Fibonacci sequence's limiting ratio. The pentagon and the decagon are the two regular polygons where this constant surfaces directly, rather than needing a longer chain of identities to reach it.
The same triangle-splitting idea generalises to any regular polygon through A = ns² ⁄ (4·tan(π⁄n)), and raising n makes the shape hug a circle ever more closely — a 100-gon already looks round to the eye. At the other extreme, the formula only holds for a regular decagon: stretch one side or skew an angle and the ten-triangle symmetry breaks, so the area then has to be found from coordinates or by summing irregular triangles instead.
- Enter your decagon's side length in the Side length field — one number, any consistent unit, since every side and angle is assumed equal.
- Area updates at once, in that same unit squared — no separate apothem or angle entry required.
- To sanity-check the result, double the Side length and confirm Area grows by four times, not two — the square of the factor, not the factor itself.
- Remember the formula assumes a regular decagon; an irregular ten-sided outline needs a different method entirely (see the FAQ below).
Worked example — a one-metre decagonal paver
A garden paver is cut as a regular decagon with every side exactly 1 metre long. Its area comes out to A = 10 × 1² ⁄ (4 × tan 18°) = 7.694208842938134 square metres, the figure to plan sealant or gravel infill around, even though the paver's outline already reads as nearly circular to the eye.
Cut the same paver at 2 metres per side instead and the area does not simply double — it becomes 30.776835371752536 square metres, four times as large. Doubling a linear side length quadruples a two-dimensional area, because each of the ten triangles has its base and its height both double, multiplying that triangle's own area by four before all ten are summed.
Questions
What is the formula for the area of a regular decagon?
A = 10s² ⁄ (4·tan 18°), where s is one side length. It comes from splitting the decagon into ten congruent isosceles triangles meeting at the centre and summing their areas; with s = 1 the formula returns 7.694208842938134, this sheet's own reference value.
Is a decagon the same shape as a dodecagon?
No — deca- means ten, dodeca- means twelve, and the names are easy to mix up. A dodecagon uses the same family of formulas with n = 12 in place of n = 10, which changes tan 18° to tan 15° and gives a different area for the same side length. The two shapes share one general formula, nothing more.
Does this formula work for an irregular ten-sided shape?
No. A = 10s² ⁄ (4·tan 18°) assumes all ten sides and all ten interior angles are equal, so one side length fully determines the shape. An irregular decagon needs its vertex coordinates plotted and its area found with the shoelace formula, or by summing irregular triangles or trapezoids cut from the outline.
How is the tan 18° in the formula derived?
It comes from bisecting the 36° angle each of the ten triangles makes at the decagon's centre, since 360° ⁄ 10 = 36°. Halving that angle gives a right triangle with an 18° angle, and tan 18° relates the half-base, s ⁄ 2, to the height, the apothem — rearranging gives apothem = s ⁄ (2·tan 18°), the term hidden inside the area formula.
What is the connection between a decagon and the golden ratio?
A regular decagon's circumradius — centre to any vertex — equals exactly φ times its side length, where φ ≈ 1.618034 is the golden ratio. That falls out of sin 18° = (√5 − 1) ⁄ 4, one of the few clean trigonometric values tied to √5, which is why the decagon and the pentagon are the regular polygons most associated with the golden ratio.
Why does doubling the side length more than double the area?
Because area is a two-dimensional quantity while side length is one-dimensional. Every one of the ten triangles has its base and height both double when s doubles, so each triangle's area is multiplied by four; summed across all ten, the whole decagon's area quadruples too — 7.694208842938134 square units at s = 1 becomes 30.776835371752536 at s = 2.