Calculating the Side Length of a Regular Polygon Given Its Apothem

The side length of a regular polygon can be calculated using the apothem, which is the distance from the center of the polygon to the midpoint of one of its sides. This article will walk you through the process of finding the side length of a regular polygon with n sides when the apothem is given as 1 unit.

Formula and Explanation

The formula to calculate the side length of a regular polygon with n sides when the apothem is 1 can be derived from the properties of isosceles triangles formed by the radii and the apothem. The formula is:

s 2tan{frac{180^circ}{n}}

This formula is derived from the relationship between the side length and the apothem in a regular polygon. Let's break it down step by step.

Step-by-Step Derivation

A regular n-gon can be divided radially into n congruent isosceles triangles. For each triangle:

Height (apothem) 1Vertex angle $frac{360^circ}{n}$Base of each isosceles triangle, which is half of the side length of the polygon $2tan{frac{180^circ}{n}}$Side of the polygon $2tan{frac{180^circ}{n}}$

Let's denote the side length as $L$, the central angle as $frac{360^circ}{n}$, and the apothem as 1. Each isosceles triangle is composed of two congruent right triangles. Therefore, the side of the polygon can be expressed as:

$L 2tan{frac{180^circ}{n}}$

From this, we can deduce that the side length $s$ of the polygon is:

$s 2tan{frac{180^circ}{n}}$

This formula can also be written using radians, where $pi 180^circ$, leading to the formula:

$s 2tan{frac{pi}{n}}$

Example Calculation

Suppose we have a regular octagon (where n 8) and we know the apothem is 1 unit. To find the side length:

$s 2tan{frac{180^circ}{8}} 2tan{22.5^circ}$

Using a calculator, we find that:

$tan{22.5^circ} approx 0.4142$

So, the side length $s approx 2 times 0.4142 0.8284$ units.

Conclusion

Understanding how to calculate the side length of a regular polygon given its apothem is a fundamental concept in geometry and has practical applications in various fields such as architecture and engineering. By mastering the formula and its derivation, you can accurately determine the dimensions of regular polygons in real-world scenarios.