Dodecagon Area Calculator

Calculate the area of a regular dodecagon using multiple methods: from side length, apothem, or circumradius. Step-by-step solutions included.

Select Method & Enter Dimensions

Result

Area of Dodecagon
279.9038
square units

Step-by-Step Solution

A = 3s^2(2 + sqrt(3))

How to Calculate Dodecagon Area

The area of a regular dodecagon (12-sided polygon) can be calculated using several methods depending on what measurement you know. The most common formula uses the side length: A = 3s2(2 + sqrt(3)), where s is the side length. This elegant formula arises because a regular dodecagon can be decomposed into 12 congruent isosceles triangles.

Area Formulas for a Regular Dodecagon

From Side Length

The most direct formula using the side length s of the dodecagon.

A = 3s2(2 + sqrt(3))

From Apothem

Using the apothem (distance from center to midpoint of a side).

A = (1/2) x P x a = 6sa

From Circumradius

Using the circumradius (distance from center to a vertex).

A = 3R2 x sin(pi/6) x 12/2

Using Triangles

Decompose into 12 triangles from center to each side.

A = 12 x (1/2) x s x a

Approximate Factor

The area constant for a regular dodecagon.

A = 11.19615... x s2

From Perimeter

Using perimeter P and apothem a together.

A = (1/2) x P x a

Deriving the Formula

The formula A = 3s2(2 + sqrt(3)) can be derived by dividing the regular dodecagon into 12 congruent isosceles triangles, each with a base equal to the side length s and height equal to the apothem. The central angle of each triangle is 360/12 = 30 degrees. Using trigonometry, the apothem equals (s/2) x cot(15 degrees), and the area of each triangle is (1/2) x s x apothem. Multiplying by 12 and simplifying yields the formula.

Numerical Constant

The factor 3(2 + sqrt(3)) equals approximately 11.19615. This means the area of a regular dodecagon is about 11.196 times the square of its side length. For comparison, a circle with the same circumradius would have area pi x R2, and the dodecagon fills about 98.86% of the circumscribed circle.

Converting Between Measurements

  • Side to Apothem: a = (s/2) x cot(pi/12) = s x (2 + sqrt(3))/2
  • Side to Circumradius: R = s / (2 x sin(pi/12))
  • Apothem to Side: s = 2a x tan(pi/12)
  • Circumradius to Side: s = 2R x sin(pi/12)

Worked Example

Find the area of a regular dodecagon with side length 10 cm:

  1. Formula: A = 3s2(2 + sqrt(3))
  2. A = 3 x (10)2 x (2 + 1.7321)
  3. A = 3 x 100 x 3.7321
  4. A = 300 x 3.7321 = 1119.62 cm2