Dodecagon Area Calculator — Three Input Methods
Calculate the area of a regular 12-sided polygon from whichever measurement you know — side length, circumradius, or apothem — with all related dimensions shown.
Known measurement
A = 3(2+√3) × s²
- 1
Square the side
5² = 25 - 2
Multiply by 3(2+√3)
3(2+√3) × 25 = 279.9038
How does this calculator work?
The area of a regular dodecagon is 3(2+√3)s² ≈ 11.196s² from side s, 3R² from circumradius R, or 12(2−√3)a² ≈ 3.215a² from apothem a. Choose the measurement you know and the calculator converts to all others.
Formula
How this is calculated
A regular dodecagon has three naturally related measurement sets: the side length s, the circumradius R (centre to vertex), and the apothem a (centre to midpoint of a side). Knowing any one of them fixes the others because the ratios are constant: R = s(√6+√2)/2 and a = s(2+√3)/2.
Each input yields a clean area formula. From side: A = 3(2+√3)s², derived by decomposing the polygon into 12 congruent triangles. From the circumradius: since A = 3(2+√3)s² and s = R(√6−√2)/2, substituting and simplifying gives the elegant A = 3R² — no surd survives. From the apothem: using s = 2a(2−√3) gives A = 12(2−√3)a² ≈ 3.215a².
All three formulae are mathematically exact; the approximate decimal multipliers are rounded to three significant figures above. The perimeter is always 12s regardless of which measurement is entered.
Frequently asked questions
The algebra works out to a rational coefficient (3) because the product of the dodecagon's geometric constants — sin(π/12) and cos(π/12) — simplifies via the identity sin(2θ) = 2sin(θ)cos(θ). For most other polygons the coefficient involves surds; the dodecagon is one of the few exceptions.
The apothem (also called the inradius) is the perpendicular distance from the centre of the polygon to any of its sides — equivalently, the radius of the largest circle that fits inside the dodecagon. For a dodecagon, a = s(2+√3)/2 ≈ 1.866s.
Yes — dodecagon comes from the Greek dōdeka (twelve) + gōnia (angle). A regular dodecagon has 12 equal sides and 12 interior angles of 150° each, for a total interior angle sum of 1800°.
Also known as
TG we-Calculate Editorial Team. (2026). Dodecagon Area Calculator — Three Input Methods [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/dodecagon-area-calculator
TG we-Calculate Editorial Team. "Dodecagon Area Calculator — Three Input Methods." TG we-Calculate. 2026. https://we-calculate.com/calculator/dodecagon-area-calculator.
TG we-Calculate Editorial Team, "Dodecagon Area Calculator — Three Input Methods," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/dodecagon-area-calculator
@misc{wecalculate_dodecagon_area_calculator, title = {Dodecagon Area Calculator — Three Input Methods}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/dodecagon-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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