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.
Count how many sides the polygon has.
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.
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.
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.
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.
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.
Area and Side Length
Count the number of sides the polygon has.
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.
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.
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.