12 Sided Polygon Calculator

Regular Dodecagon Formulas:

\[ s = 2r \sin\left(\frac{180°}{12}\right) \] \[ \text{Perimeter} = 12s \] \[ \text{Area} = 3r^2 \sqrt{3} \]

Side Length (s):

Circumradius (r):

Perimeter:

Area:

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1. What is a Regular Dodecagon?

A regular dodecagon is a 12-sided polygon where all sides are equal in length and all angles are equal in measure (150° each). It has twelve lines of symmetry and rotational symmetry of order 12.

2. Key Formulas

The calculator uses these fundamental formulas:

\[ s = 2r \sin\left(\frac{π}{12}\right) \] \[ \text{Perimeter} = 12s \] \[ \text{Area} = 3r^2 \sqrt{3} \]

Where:

\( s \) — Side length
\( r \) — Circumradius (distance from center to vertex)
\( π \) — Pi (approximately 3.14159)

3. Properties of Dodecagon

Details: In a regular dodecagon:

Each internal angle measures 150°
Each external angle measures 30°
The sum of all interior angles is 1800°
It can be divided into 12 isosceles triangles

4. Using the Calculator

Tips: Enter either the side length or circumradius. The calculator will compute all other properties. Values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between side length and circumradius?
A: Side length is the length of one edge, while circumradius is the distance from center to any vertex.

Q2: How is the area formula derived?
A: The area comes from summing 12 isosceles triangles (each with area \( \frac{1}{2}r^2 \sin(30°) \)).

Q3: Can I calculate the apothem with this?
A: Yes, apothem \( a = r \cos(15°) \), which is about 0.9659 times the circumradius.

Q4: What are real-world examples of dodecagons?
A: Some coins, architectural elements, and game boards use dodecagonal shapes.

Q5: How precise are the calculations?
A: Results are accurate to 4 decimal places, though exact values would keep \( \sqrt{3} \) symbolic.