Math Antics - Perimeter

# How to Find the Perimeter of a Polygon

In geometry, the term "perimeter" usually refers to thetotal distance around the outside of a polygon, a 2-dimensional shape of 3 or more sides and angles. In other words, a given shape's perimeter is the sum of the lengths of all of its sides. The ease with which a shape's perimeter can be found depends on several factors - namely, whether the polygon is aregularpolygon (a polygon whose sides and angles are all equal), and, if not, whether all the side lengths in the shape are known. If neither of these conditions are true, the difficulty of finding the shape's perimeter depends on how much information is known about the shape.

## Steps

1. Find and add the lengths of all the polygon's sides.The perimeter ofanypolygon 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 preventing you from finding the perimeter of the shape itself.
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.
3. Multiply a regular polygon's side length by the number of sides.Polygons whose sides areallequal in length and whose angles areallthe 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), or16 inches (40.6 cm), while the perimeter of an equilateral triangle with a side length of 4 inches (10.2 cm) is 4 × 3, or12 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.
4. Alternatively, use the area andapothemof a regular polygon to find its perimeter.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 theapothem, 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)/2allows 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 inches2and 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 is16 inches (40.6 cm)- this is the same answer that we got above with the standard method.

## Community Q&A

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• Question
The length of the sides of a pentagon are consecutive odd integers. What is the perimeter if the longest side is 15?
The four consecutive odd integers just below 15 are 7, 9, 11 and 13, so the perimeter is 7 + 9 + 11 + 13 + 15 = 55.
Thanks!
• Question
I have a regular octagon with an area of 55cm^2. How do I find the side length if the perimeter is not given?
The area of a regular octagon is equal to the square of the side, multiplied by 4.828. A = (4.828)(a²). Therefore, a² = A / 4.828, and a = √(A/4.828) = √(.207)(A). Use 55 for A, and calculate a.
Thanks!
• Question
What is the area of a hexagon with all of the sides being 10 cm?
The area formula for a regular hexagon is [(3√3) / 2] s², where s is the length of a side. So in this case it's (2.598)(10)² = 259.8 square centimeters.
Thanks!
• Question
How do I find the perimeter of a pentagon with sides of lengths 12m, 50m, 88m, 125m and 100m?
Thanks!
• Question
If length and width isn't given, what do I do?
wikiHow Contributor
Are the angles given? If they are you can split the polygon into triangles and use the sine and cosine rules to find the sides.
Thanks!
• Question
The sides of a polygon have length of 5 cm, 3 cm, 7 cm, 10 cm and 12 cm. The perimeter of a similar polygon is 148 cm. What is the length of the smaller side of the other polygon?
The first perimeter is 5 + 3 + 7 + 10 + 12 = 37 cm. The ratio of the first perimeter to the second perimeter is 37:148. The ratio of the first polygon's smallest side to the second polygon's smallest side is also 37:148. Let x be the length of the second polygon's smallest side. Then 3 / x = 37 / 148. By cross-multiplying, 37x = 444. So x = 12 cm.
Thanks!
• Question
How do I find the perimeter of a polygon when the length of one side is unknown?
As your question is posed, the perimeter cannot be found. You would have to know some angles.
Thanks!
• Question
What is the perimeter of the polygon 12.7 by 8 by 11.4 by 7.3?
12.7 + 8 + 11.4 + 7.3 = 39.4.
Thanks!
• Question
What is the perimeter of a pentagon if each side is 90 cm?
Multiply 90 cm by 5.
Thanks!
200 characters left
• Although we think of a polygon in terms of how many sides it has, the word "polygon" actually means "multiple angles."

## Things You'll Need

• Ruler or measuring stick
• Pencil and paper or calculator

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Date: 15.12.2018, 04:37 / Views: 91245