How To Calculate The Apothem Of A Polygon

A polygon is a shape that has any number of straight sides, such as a triangle, square or hexagon. The apothem refers to the length of the line the connects the center of a regular polygon to the midpoint of any of the sides. A regular polygon has all congruent sides; if the polygon is irregular, there is not a midpoint equidistant from the midpoint of all sides. You can calculate the apothem if you know the area. If you know the area and the side lengths, you can use a simpler formula.

Step 1

Count how many sides the polygon has.

Step 2

Divide the area of the polygon by the number of sides the polygon has. For example, if the area of a square is 36, you would divide 36 by 4 and get 9.

Step 3

Divide pi by the number of sides in the polygon. In this example, you would divide pi, about 3.14, by 4, the number of sides in a square, to get 0.785.

Step 4

Use your scientific calculator to calculate the tangent of the result from Step 3 in radians. If you have your calculator set to degrees you will get an incorrect result. In this example, the tangent of 0.785 equals about 1.0.

Step 5

Divide the result from Step 2 by the result from Step 4. Continuing the example, you would divide 9 by 1 and get about 9. In the case of a square, this step may seem superfluous, but it is necessary, especially for many-sided polygons.

Step 6

Find the apothem length by taking the square root of the result from Step 5. Completing the example, the square root of 9 equals 3, so the length of the apothem equals 3.

Step 1

Count the number of sides the polygon has.

Step 2

Multiply the number of sides times the length of one side to calculate the perimeter. For example, if you have a hexagon with each side measuring 7 inches, the perimeter would be 42 inches.

Step 3

Multiply the area of the hexagon by 2. In this example, the area equals 127.31 so you would double that to get 254.62.

Step 4

Divide the result from Step 3 by the perimeter, found in Step 2, to calculate the apothem. Concluding this example, you would divide 254.62 by 42 to find the length of the apothem equals about 6.06 inches.