How to calculate the perimeter and area of an octagon

The geometric figure of eight sides, called octagon or octagon, is usually represented in two dimensions as a drawing or a flat object, a common example being a traffic signal. The area of an octagonal figure is easily calculated with basic mathematics. Calculating the side, sides or perimeter of an octagon, is a simple matter of adding the lengths of the sides. Although rare, three-dimensional objects can also be formed with eight sides and the lateral area is calculated with the same formula as a square or rectangle. In we want to make it easy for you and we explain how to calculate the perimeter and area of an octagon.

You will need to:

Ruler
Calculator

Steps to follow:

one

The first thing you have to do is measure the length of each side of the octagon ; It should be noted that this polygon can be regular, that is, that all its sides are identical and measure the same, or irregular in case the sides are different.

two

To know the perimeter of a regular octagon -like the one you see in the drawing below-, you must multiply the length of one side of the octagon by the number of sides that in the octagon is 8. So, the mathematical formula says that P = l · 8

For example, if the eight sides of the octagon have an identical length of five centimeters, the perimeter of the octagon is calculated:

5 cm x 8 sides = 40 cm perimeter

3

In the case of irregular octagons, you must determine the perimeter by calculating each side separately and the sum of these figures .

For example: if the first side is 5 centimeters, the second side is 4 centimeters, the third side is 7 centimeters, the fourth is 3 centimeters and the sides five, six, seven and eight are 10 centimeters, the perimeter of the octagon would be equal to 60 centimeters

Perimeter = 5 + 4 + 7 + 3 + 10 + 10 + 10 + 10 = 60 cm.

4

If we want to calculate the surface or area of a regular octagon, we must apply the mathematical formula that says that the area is equal to the multiplication of the perimeter by the apothem divided by two.

So, we already know how to calculate the perimeter of an octagon, but what is the apothem ? It is the distance that separates the center of the polygon from the center point of each side of the octagon; If you look at the image, we have indicated it in green.

Following the example, if each side is 5 cm and the apothem is 10 cm, we calculate the surface of the octagon by multiplying the side by 8 and by the apothem and dividing the result by two:

S = (5 cm · 8 cm) · 10/2 = 40 · 10/2 = 200 cm²

5

Another equally valid option for calculating the surface of a regular octagon is to divide the polygon into eight equal triangles, calculate its area and then multiply it by eight. In this way, the apothem of the regular octagon will be equal to the height of each of these triangles and the side equal to the base, which are the two elements we need to calculate the area of a triangle.

Thus, the surface of a triangle is obtained by applying the formula that says that it is equal to the multiplication of the base by the height and dividing its result by two:

S = (5 · 10) / 2 = 50/2 = 25 cm²

Once this is done, we will only need to multiply the surface or area of the triangle by 8, which is the number of regular triangles that make up the polygon with eight sides:

S = 25 · 8 = = 200 cm²

As we see, the result is the same despite applying two different methods.