Regular N-gon Calculator (Area, Apothem, Radius)
Solve for all geometric properties of any regular polygon. Input the **Number of Sides ($n$)** and **one** known dimension (Side Length $s$, Apothem $a$, or Radius $R$).
Input (Minimum $n$ and one dimension)
Key Results
Perimeter ($P$)
Apothem ($a$)
Radius ($R$)
Area ($A$)
Angle Measures (Degrees)
Internal Angle ($\theta_i$)
Central Angle ($\theta_c$)
Exterior Angle ($\theta_e$)
Step-by-Step Solution
The Fundamental Triangle and Trigonometry
The key to solving any regular N-gon is to divide it into $n$ congruent **isosceles triangles**. Each triangle is formed by two radii ($R$) and one side ($s$). Bisecting this triangle creates a **right triangle**, known as the fundamental triangle, with the following properties:
- **Leg 1:** Apothem ($a$).
- **Leg 2:** Half of the side length ($s/2$).
- **Hypotenuse:** Radius ($R$).
- **Angle at Center:** Half of the central angle ($\frac{180^\circ}{n}$).
Using this right triangle and the tangent function, we establish the core relationship:
Key Formulas for a Regular Polygon
Once one property (like the apothem) is known, all other properties can be calculated.
| Property | Formula |
|---|---|
| Perimeter ($P$) | $P = n \cdot s$ |
| Area ($A$) | $$A = \frac{1}{2} a P \quad \text{or} \quad A = \frac{1}{4} n s^2 \cot\left(\frac{180^\circ}{n}\right)$$ |
| Internal Angle ($\theta_i$) | $$\theta_i = \frac{(n-2) \times 180^\circ}{n}$$ |
| Central Angle ($\theta_c$) | $$\theta_c = \frac{360^\circ}{n}$$ |