Calculate the area, perimeter, apothem, circumradius, interior/exterior angles, and number of diagonals of a regular 12-sided polygon (dodecagon) from side length, circumradius, or apothem.
A regular dodecagon is a twelve-sided polygon with all sides and angles equal. It is one of the most symmetrical plane figures, appearing in architecture (clock faces, coin designs), tiling patterns, and mathematical art. The dodecagon has 12 lines of symmetry and rotational symmetry of order 12.
The area of a regular dodecagon with side length s is A = 3s²(2 + √3), approximately 11.196 × s². This formula comes from dividing the dodecagon into 12 congruent isosceles triangles from the center and summing their areas. Equivalently, A = ½ × perimeter × apothem.
Each interior angle of a regular dodecagon is 150° and each exterior angle is 30°. These values follow from the general formulas for regular n-gons: interior angle = (n − 2) × 180° / n and exterior angle = 360° / n. The dodecagon has 54 diagonals, calculated as n(n − 3)/2 = 12 × 9/2.
The apothem (distance from center to side midpoint) and circumradius (distance from center to vertex) are related to the side length by a = s / (2 tan π/12) and R = s / (2 sin π/12). As the number of sides increases, regular polygons approximate a circle — the dodecagon already captures about 98.9% of its circumscribed circle's area.
This calculator lets you compute all dodecagon properties from any one known measurement — side length, circumradius, or apothem. It includes presets, visual comparison bars, a reference table, and a comparison with other regular polygons from triangles through dodecagons.
The Regular Dodecagon Area & Properties Calculator is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Area, Perimeter, Side Length in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Area: A = 3s²(2 + √3) ≈ 11.196 × s² Perimeter: P = 12s Apothem: a = s / (2 × tan(π/12)) Circumradius: R = s / (2 × sin(π/12)) Interior Angle: (n − 2) × 180° / n = 150° Exterior Angle: 360° / n = 30° Diagonals: n(n − 3) / 2 = 54
Result: Area ≈ 1,119.62 cm², Perimeter = 120 cm, Apothem ≈ 18.66 cm, R ≈ 19.32 cm
For a regular dodecagon with s = 10 cm: Area = 3 × 100 × (2 + √3) = 300 × 3.732 ≈ 1,119.62 cm². Perimeter = 12 × 10 = 120 cm. Apothem = 10 / (2 × tan 15°) ≈ 18.66 cm. Circumradius = 10 / (2 × sin 15°) ≈ 19.32 cm.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Regular Dodecagon Area & Properties Calculator calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
The area is A = 3s²(2 + √3) ≈ 11.196 × s², where s is the side length. Equivalently, A = ½ × perimeter × apothem.
A regular dodecagon has 54 diagonals, calculated as n(n − 3)/2 = 12 × 9 / 2 = 54. Use this as a practical reminder before finalizing the result.
Each interior angle is 150° and each exterior angle is 30°. The sum of all interior angles is 1800°.
A regular dodecagon inscribed in a circle of radius R covers about 98.9% of the circle's area, making it an excellent circular approximation.
The apothem is the distance from the center to the midpoint of a side (perpendicular). The circumradius is from the center to a vertex. The circumradius is always larger: R = a / cos(π/n).
Clock faces (12 hour marks), the British threepence coin, some architectural designs, stop sign intersections, and mathematical tiling patterns. Keep this note short and outcome-focused for reuse.