Free Dodecagon Calculator

What Is a Dodecagon?

A dodecagon is a polygon with twelve straight sides, twelve vertices, and twelve interior angles. The name derives from the Greek dodeka (twelve) and gonia (angle), just as pentagon (five) or octagon (eight) follow the same pattern. When all sides are equal in length and every interior angle has the same measure, the shape is called a regular dodecagon. This regular dodecagon calculator is built to help you quickly find any unknown dimension — side length, area, perimeter, diagonal, incircle radius, or circumcircle radius — as long as you supply one known value.

Key Properties of a Regular Dodecagon

• Interior angle: Each interior angle of a regular dodecagon measures exactly $150^{\circ}$. Because all twelve are equal, the sum of the interior angles is $12 \times 150^{\circ} = 1800^{\circ}$.
• Exterior angle: The exterior angle is the supplement of the interior angle, so $\beta = 180^{\circ} - 150^{\circ} = 30^{\circ}$. This angle does not change with the size of the polygon.
• Number of diagonals: A dodecagon has a total of 54 diagonals. The general formula for an $n$-sided polygon is $\frac{n(n-3)}{2}$; substituting $n = 12$ gives $\frac{12 \times 9}{2} = 54$. (Alternatively, choose any two vertices – $\binom{12}{2} = 66$ – and subtract the 12 sides.)
• Sum of interior angles: Using the standard formula $(n-2) \times 180^{\circ}$ with $n = 12$ yields the same $1800^{\circ}$ result.

Essential Formulas for a Regular Dodecagon

Let the side length be denoted by $a$. All other measures can be expressed in terms of $a$ using the following relationships:

• Perimeter: $P = 12a$.
• Area: $\displaystyle \text{Area} = 3 (2 + \sqrt{3}) a^{2} \approx 11.196 a^{2}$.
• Circumradius (radius of the circle that passes through all vertices): $\displaystyle R = \frac{1}{2} a (\sqrt{6} + \sqrt{2}) \approx 1.932 a$.
• Inradius (radius of the circle inscribed inside the dodecagon, tangent to each side): $\displaystyle r = \frac{1}{2} a (2 + \sqrt{3}) \approx 1.866 a$.

These formulas are exact; the decimal approximations are convenient for quick estimates.

How to Use This Dodecagon Calculator

The tool works with a single piece of input. Depending on what you already know, you may enter:

• One side length (then the calculator determines the area, perimeter, both radii, and the number of diagonals);
• The perimeter;
• The total area;
• The length of one diagonal;
• The radius of the incircle or the circumcircle.

Once you type your known value, all remaining parameters are displayed immediately. You can also check the Try other regular polygons box below the diagram to change the number of sides and explore shapes like pentagons, hexagons, or decagons using the same interface. Whether you are a student studying geometry, a teacher preparing materials, or a hobbyist working on a design, this 12‑sided polygon calculator saves time and ensures accuracy.