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 $R$).

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.

Frequently Asked Questions (FAQ)

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\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{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)

\theta_i = \frac{(n-2) \times 180^\circ}{n}

\theta_c = \frac{360^\circ}{n}

A = \frac{1}{2} a P

$\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{a}$