How this instrument works
A regular dodecagon has twelve equal sides and twelve equal interior angles, each measuring exactly 150°. Its area follows the general regular-polygon rule A = n·s² ⁄ (4·tan(π⁄n)) with n fixed at 12, but the dodecagon is one of the few n-gons where that expression collapses to a clean closed form: tan(π⁄12), or tan(15°), simplifies exactly to the surd 2 − √3, so the whole formula reduces to A = 3(2 + √3)·s² — no trigonometric table needed once you know that one identity.
The formula comes from splitting the shape into twelve congruent isosceles triangles that meet at the center, each spanning a 30° wedge with base s. That is the same construction behind every regular polygon's area rule; the dodecagon just happens to land on tan(15°), a tangent value that a hexagon (n = 6, giving tan30° = 1⁄√3) or a decagon (n = 10) do not share in such a tidy form. A regular dodecagon inscribed in the same circle as a regular hexagon, in fact, has its extra six vertices sitting exactly at the midpoint of each of the hexagon's six arcs.
One edge case worth noticing: holding the side length fixed and increasing n does not add area in proportion to the extra sides. Because tan(π⁄n) shrinks roughly like π⁄n as n grows, the area term behaves closer to n² than to n — a dodecagon has roughly 4.3 times a same-sided hexagon's area, not twice, even though it has only twice the sides. Adding sides at a fixed side length mostly adds area, not just more edges.
- Enter the length of one edge into the Side length field — all twelve sides of a regular dodecagon are equal, so a single measurement is all the formula needs.
- Read the result in Area (regular dodecagon); it updates as soon as Side length changes, computed from A = 3(2 + √3)·s² behind the scenes.
- Keep Side length and the reported area in matching units — a side entered in centimeters returns an area in square centimeters, not square meters.
- When measuring a real object, take the length of one flat edge, not the distance across the shape — a dodecagon has several different diagonal lengths, and none of them equals s.
Worked example — a dodecagon with 3-unit sides
Set Side length to 3. The engine finds A = 12 × 3² ⁄ (4 × tan(π⁄12)) = 108 ⁄ (4 × 0.267949192…) = 108 ⁄ 1.071796770… ≈ 100.765372, and Area (regular dodecagon) reports 100.76537180435969 to full precision, not an early-rounded approximation carried forward.
The closed form reaches the identical figure by a shorter route: A = 3(2 + √3) × 3² = 3(2 + √3) × 9 = 27(2 + √3) = 100.76537180435969. Because area scales with the square of the side, tripling a unit-side dodecagon's area of 3(2 + √3) ≈ 11.196152 by 3² = 9 lands on exactly that same number, not merely close to it.
Questions
What is the area formula for a regular dodecagon?
A = n·s² ⁄ (4·tan(π⁄n)) specialized to n = 12, or equivalently the closed form A = 3(2 + √3)·s² once tan(15°) is replaced by its exact surd value, 2 − √3. Both give the same number for any side length; the second skips the trigonometry and can be checked by hand with only a square root of 3.
Why does tan(π⁄12) simplify to 2 − √3?
π⁄12 radians is 15°, and 15° = 45° − 30°, two angles with well-known tangents. The tangent subtraction identity gives tan(45° − 30°) = (1 − tan30°) ⁄ (1 + tan30°); clearing the surds in that fraction leaves exactly 2 − √3 ≈ 0.267949, with no rounding until the final decimal display.
How does this relate to working out a hexagon's area?
A regular hexagon (n = 6) shares the same skeleton, A = n·s² ⁄ (4·tan(π⁄n)), only with tan30° = 1⁄√3 in place of tan15°. Inscribe both in one circle and the dodecagon's twelve vertices interleave the hexagon's six, one landing at the midpoint of each hexagon arc.
What's the most common mistake when measuring the side length?
Measuring the distance across the shape instead of one flat edge. A dodecagon has several distinct diagonal lengths depending on which vertices are joined, and none of them equals s in the formula — only the straight segment between two adjacent vertices does.
Does doubling the number of sides double the area at a fixed side length?
No. Because tan(π⁄n) shrinks roughly like π⁄n as n grows, the area term behaves closer to n² than to n. A dodecagon (n = 12) has roughly 4.3 times a hexagon's area (n = 6) at the same side length, not twice — most of that growth is area, not merely extra perimeter.
Can regular dodecagons tile a flat surface by themselves?
No. Each interior angle is 150°, and 360° is not a whole multiple of 150° (360 ⁄ 150 = 2.4), so dodecagons alone leave gaps. They do appear in mixed tessellations — one classical pattern alternates a dodecagon with a hexagon and a square meeting at every vertex.