Math & Conversions Regular N-gon Calculator 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) Number of Sides ($n$) Side Length ($s$) Apothem ($a$) Radius ($R$) Calculate Polygon Properties 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: $$\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{a}$$ 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 ($\theta_c$) $$\theta_c = \frac{360^\circ}{n}$$ Frequently Asked Questions (FAQ) What is a regular N-gon? A regular N-gon (regular polygon) is a closed, two-dimensional shape with 'N' number of equal sides and 'N' equal interior angles. Examples include the square (N=4), pentagon (N=5), and hexagon (N=6). What is the apothem? The apothem ($a$) is the distance from the center of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle and is used as the height when calculating the area of the polygon's fundamental triangle. What is the relationship between the central angle and the number of sides? The central angle ($\theta_c$) is the angle formed by two radii drawn to consecutive vertices. It is calculated by dividing the full circle ($360^\circ$) by the number of sides ($n$): $\theta_c = 360^\circ

Subcategories in Math & Conversions Regular N-gon Calculator 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) Number of Sides ($n$) Side Length ($s$) Apothem ($a$) Radius ($R$) Calculate Polygon Properties 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: $$\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{a}$$ 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 ($\theta_c$) $$\theta_c = \frac{360^\circ}{n}$$ Frequently Asked Questions (FAQ) What is a regular N-gon? A regular N-gon (regular polygon) is a closed, two-dimensional shape with 'N' number of equal sides and 'N' equal interior angles. Examples include the square (N=4), pentagon (N=5), and hexagon (N=6). What is the apothem? The apothem ($a$) is the distance from the center of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle and is used as the height when calculating the area of the polygon's fundamental triangle. What is the relationship between the central angle and the number of sides? The central angle ($\theta_c$) is the angle formed by two radii drawn to consecutive vertices. It is calculated by dividing the full circle ($360^\circ$) by the number of sides ($n$): $\theta_c = 360^\circ.