Dodecagon Calculator

What is a Dodecagon?

A dodecagon is a twelve-sided polygon in plane geometry. When all twelve sides and all twelve interior angles are congruent, it is classified as a regular dodecagon. Regular dodecagons possess rich geometric symmetry (dihedral symmetry $D_{12}$) and appear frequently in architectural tiling patterns, clock faces, coins, and engineering designs.

Key Geometric Formulas for a Regular Dodecagon

For a regular dodecagon with side length $a$:

1. Perimeter

The perimeter is simply the sum of all 12 equal side lengths:

$$P = 12a$$

2. Area

The exact formula for the area of a regular dodecagon derived from trigonometry is:

$$A = 3 \cot\left(\frac{\pi}{12}\right) a^2 = 3(2 + \sqrt{3}) a^2 \approx 11.1961524 \cdot a^2$$

Alternatively, if you know the circumradius $R$ or inradius (apothem) $r$:

$$A = 3 R^2$$

$$A = 12(2 - \sqrt{3}) r^2 \approx 3.21539 r^2$$

3. Inradius (Apothem) and Circumradius

The inradius $r$ (the radius of the inscribed circle, also called the apothem) is:

$$r = \frac{a}{2 \tan(15^\circ)} = \frac{a}{2}(2 + \sqrt{3}) \approx 1.866025 \cdot a$$

The circumradius $R$ (the radius of the circle passing through all 12 vertices) is:

$$R = \frac{a}{2 \sin(15^\circ)} = \frac{a}{2}(\sqrt{6} + \sqrt{2}) \approx 1.931852 \cdot a$$

4. Angles and Diagonals

• Interior angle: $\alpha = \frac{(12 - 2) \times 180^\circ}{12} = 150^\circ = \frac{5\pi}{6}\text{ rad}$
• Exterior angle: $\beta = \frac{360^\circ}{12} = 30^\circ = \frac{\pi}{6}\text{ rad}$
• Sum of interior angles: $(12 - 2) \times 180^\circ = 1800^\circ$
• Total number of diagonals: $\frac{n(n-3)}{2} = \frac{12 \times 9}{2} = 54\text{ diagonals}$

Summary of Dodecagon Diagonals

From any single vertex of a regular dodecagon, there are 9 diagonals spanning different numbers of edges:

• $d_2$ (spanning 2 edges): $d_2 = R = \frac{a}{2}(\sqrt{6}+\sqrt{2})$
• $d_3$ (spanning 3 edges): $d_3 = R\sqrt{2} = a(\sqrt{3}+1)$
• $d_4$ (spanning 4 edges): $d_4 = R\sqrt{3}$
• $d_5$ (spanning 5 edges): $d_5 = 2r = a(2+\sqrt{3})$
• $d_6$ (main diameter across opposite vertices): $d_6 = 2R = a(\sqrt{6}+\sqrt{2})$

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