How this instrument works
A regular dodecagon is a twelve-sided polygon with every side and every interior angle equal — each interior angle works out to exactly 150°, from (n − 2) × 180° ⁄ n with n = 12, giving 1800° ⁄ 12. Slice the shape from its center and it falls apart into twelve identical isosceles triangles, each with a 30° angle at the apex, since the full 360° at the center splits evenly twelve ways. Sum the area of those twelve triangles — half the base times the apothem, over and over — and the formula A = 12s² ⁄ (4·tan(15°)) is exactly what falls out, with 15° as half of each triangle's 30° apex angle.
Because tan(15°) has an exact closed form — tan(15°) = 2 − √3 — the whole expression collapses further, to A = (6 + 3√3)s², with no trigonometry left visible at all. For a side of 1 that reads 6 + 3 × 1.7320508… ≈ 11.19615, matching the formula above digit for digit. That tidiness is not universal: a heptagon or an 11-gon has no comparable closed form for its tangent, so those areas can only be approximated numerically. Twelve happens to land on one of the angle counts where nested square roots are enough.
There is a neater fact hiding one layer down, tied to the circle around the shape rather than its side. A dodecagon inscribed in a circle of radius R has area exactly 3R² — no π, no square root, just three times the radius squared — because splitting it into twelve triangles from the center again gives each one area (1/2)R²sin(30°), and sin(30°) is exactly 1/2. Scaling runs the other way too: halve the side length and the area drops to a quarter, since it follows the square of a linear measurement rather than the length itself.
- Enter the length of one side of the regular dodecagon into the Side length field, in any unit you like.
- Read Area — the calculator returns it in that same unit, squared, worked from the formula above.
- Check the scaling if you like: doubling the entry in Side length should return four times the Area, not twice, since area follows the square of a length.
- Side length must stay above zero; the field enforces a small positive minimum so the tangent in the formula never has to divide by nothing.
Worked example — a dodecagon with a 1-unit side
Set Side length to 1. Twelve times 1² is 12 on top; four times tan(15°) is 4 × 0.267949… ≈ 1.071797 on the bottom; dividing the two gives 12 ⁄ 1.071797 ≈ 11.19615, matching the Area field's fuller reading of 11.196152422706632 exactly.
The exact closed form checks the same figure a second way, with no tangent involved: 6 + 3√3 = 6 + 3 × 1.7320508075688772… = 11.196152422706632 — the identical number, digit for digit, confirming both routes to the formula agree completely.
Try Side length = 2 next and Area jumps to 44.78460969082653 — four times the s = 1 result, not twice, because area follows the square of a linear measurement no matter how many sides the polygon has.
Questions
What is the formula for the area of a regular dodecagon?
A = 12s² ⁄ (4·tan(15°)), where s is the length of one side — the general regular-polygon formula A = ns² ⁄ (4·tan(π⁄n)) with n = 12. It simplifies to the exact closed form A = (6 + 3√3)s², since tan(15°) has the known value 2 − √3.
Why does the formula use tan(15°) instead of some other angle?
15° is π⁄12, half of the 30° angle each of the twelve triangles makes at the center when the dodecagon is split from its middle. Halving that central angle and taking its tangent is how the apothem — the triangle's height — is recovered from half the base, s⁄2.
How do I find the area of a dodecagon knowing only its circumradius, not its side?
Use A = 3R², where R is the distance from the center to a vertex — a dodecagon inscribed in a circle of radius R always has exactly three times that radius squared as its area, because sin(30°) reduces to exactly 1⁄2. Convert to a side length first with s = 2R sin(15°) to use this calculator's Side length field instead.
What is a common mistake when using this formula?
Doubling the side length and expecting the area to double. Area scales with the square of a linear measurement, so a doubled side quadruples it — going from s = 1 (area ≈ 11.1962) to s = 2 gives 44.7846, four times as much, not two.
Does the formula change for an irregular dodecagon?
Yes — A = 12s² ⁄ (4·tan(15°)) only holds when all twelve sides and all twelve angles are equal. An irregular twelve-sided figure has to be split into triangles or trapezoids and measured piece by piece, or computed from its vertex coordinates with the shoelace formula instead.
How many sides does a dodecagon have, and what is its interior angle?
Twelve sides and twelve equal interior angles of 150° each, found from (n − 2) × 180° ⁄ n with n = 12, giving 1800° ⁄ 12 = 150°. That leaves a 30° angle at the center of each of the twelve triangles the area formula is built from.