How to Find the Perimeter of a Polygon: A Comprehensive Guide

Introduction to Polygons

Polygons are two-dimensional (2D) shapes formed by multiple straight-line segments, known as sides or edges. The point where two sides meet is called a vertex (plural: vertices). The perimeter of a polygon is the total length around the shape, which is the sum of the lengths of all its sides.

Steps to Find the Perimeter of a Polygon

To find the perimeter of a polygon, you need to follow these steps:

Identify the Sides: Determine the lengths of all the sides of the polygon. Add the Lengths: Sum the lengths of all the sides together.

The formula for the perimeter P of a polygon with n sides is:

P s_1 s_2 s_3 ldots s_n

where s_1, s_2, s_3, ldots, s_n are the lengths of the sides.

Examples of Finding Perimeter

Example 1: For a triangle with sides of lengths 3, 4, and 5:

P 3 4 5 12

Example 2: For a rectangle with length 5 and width 3:

P 2 × length width 2 × 5 3 2 × 8 16

Special Cases: Regular Polygons

A regular polygon has all sides and angles equal. The perimeter of a regular polygon is calculated as:

P n × s

where n is the number of sides and s is the length of one side.

Advanced Case: Regular Convex n-gon Inscribed in a Circle

In this more advanced case, we can generalize to a regular convex n-gon inscribed in a circle with an arbitrary radius r. Let IE be the radius and IK be the apothem (the perpendicular distance from the center of the polygon to one of its sides). The area of the polygon can be given by:

A ? × a × s × n

where a is the apothem, s is the side length, and n is the number of sides.

To find the perimeter of this polygon, we use:

1. The measure of a circular angle (the measure of an angle within a convex regular polygon with sides situated on consecutive vertices) is 360° / n.

2. The apothem a is an angle bisector and a perpendicular bisector, so each individual angle angle KIDangle KIC180°/n.

3. If ID r, then DK rsin(180°/n).

4. The length of DC is 2rsin(180°/n).

5. Therefore, the total perimeter is 2nrsin(180°/n) or in radians, P 2nrsin(π/n).