How to Find the Perimeter of a Polygon

Step-by-Step Guide

1. 1

   Step 1: Find and add the lengths of all the polygon's sides.

   The perimeter of any polygon can be calculated by finding the length of each side individually, then adding all of these lengths together.
   
   This is the most straightforward way to find a polygon's perimeter and, in shapes where no two of the sides are equal, it is usually the only accurate way to do so.
   
   As a simple example, an irregular polygon with side lengths of 5, 5, 4, 3, and 3, would have a perimeter of 5 + 5 + 4 + 3 + 3 = 20 If one or more of the side lengths in your polygon is unknown, the process of calculating the perimeter can become difficult and may require more advanced knowledge of geometry.
   
   For instance, if your polygon is a right triangle (or can be divided into right triangles, trigonometry can be a useful tool for finding the lengths of unknown sides which are are preventing you from finding the perimeter of the shape itself.
2. 2

   Step 2: Multiply the lengths of equal sides by the number of equal sides.

   Certain types of polygons have two or more equal sides.
   
   For instance, isosceles triangles and isosceles trapezoids have 2 of the sides of the same length, while parallelograms and rectangles have 2 pairs of opposing sides with equal length.
   
   In these cases, when if you know the length of one of the identical sides, you can multiply this length by the number of sides which share this length, then add the lengths of any unequal sides to find the perimeter of the overall shape.
   
   For example, let's consider an isosceles triangle that has two sides with a length of 5 inches (12.7 cm) and one side with a length of 4 inches (10.2 cm).
   
   Here, to find the perimeter, we would take the length of the equal sides (5) and multiply it by the number of equal sides (2), then add the lengths of the remaining unequal side. (5 × 2) + 4 = 10 + 4 = 14 inches (35.6 cm).
   
   As an example of a shape with multiple pairs of equal sides, let's consider a parallelogram with 2 sides with a length of 5 inches (12.7 cm) and 2 with a length of
   4.
   
   To find the perimeter, we would multiply the length of the longer side by 2 and the length of the shorter side by, then add the products together. (2 × 5) + (2 × 4) = 10 + 8 = 18 inches (45.7 cm).
   
   Note that this method can also be used for squares and rhombuses, which, along with rectangles, are special cases of parallelograms. , Polygons whose sides are all equal in length and whose angles are all the same size are called regular polygons.
   
   For example, squares and equilateral triangles are regular polygons, as are perfect pentagons (as exemplified by the Chrysler logo) and octagons (as exemplified by stop signs).
   
   If a shape is a regular polygon, finding its perimeter is a simple matter of multiplying the length of one side by the number of sides in the shape.
   
   For example, the perimeter of a perfect square with a side length of 4 inches (10.2 cm) is 4 × 4 (because a square has 4 sides), or 16 inches (40.6 cm), while the perimeter of an equilateral triangle with a side length of 4 inches (10.2 cm) is 4 × 3, or 12 inches (30.5 cm).
   
   This same basic process also works for non-regular polygons whose sides all have equal lengths.
   
   For instance, although a rhombus is not a regular polygon because its angles are not all the same size, you can find its perimeter by multiplying the length of one side by the number of sides because all 4 of its sides are the same length. , Though simply multiplying the length of one of a regular polygon's sides by the number of sides in the polygon is the easiest way to find its perimeter, it's not the only way.
   
   The distance from the center of the polygon to the exact middle of one of its sides, called the apothem, is part of an equation that allows you to find its perimeter, provided you also know the polygon's area.
   
   Inserting known values for area and apothem into the equation (Area) = (Perimeter) × (Apothem)/2 allows you to solve for the polygon's area using simple algebra.
   
   For example, the square with a side length of 4 inches (10.2 cm) in the above example has an area of 16 inches2 and an apothem of 2 inches (5.1 cm).
   
   Using our new equation, we solve for perimeter as follows: 16 = (perimeter) × 2/2 16 = (perimeter) × 1 16 = perimeter.
   
   The square's perimeter is 16 inches (40.6 cm)
   - this is the same answer that we got above with the standard method.
3. 3

   Step 3: Multiply a regular polygon's side length by the number of sides.

4. 4

   Step 4: Alternatively

5. 5

   Step 5: use the area and apothem of a regular polygon to find its perimeter.

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