How to Find the Apothem Given the Perimeter

Only regular polygons, like the Pentagon, have apothems.

A polygon's apothem measures the distance from the shape's center to the center of one of its sides. Because the apothem has to be the same length for each of the sides, only regular polygons -- polygons with all sides equal in length and all interior angles equal in degrees -- may possess them. The apothem is tied to the polygon's area by the equation area = (apothem x perimeter)/2. By knowing the polygon's area along with its perimeter, you can work the equation backwards and find the shape's apothem.

Things You'll Need

• Calculator

Instructions

1. Write down the regular polygon's area. For this example, let the polygon be a regular pentagon, which has five sides, with an area of 40 square inches.

2. Multiply the regular polygon's area by 2. Multiplying 40 square inches by 2 equals 80 square inches.

3. Divide the doubled area by the polygon's perimeter. For this example, let the perimeter be 24 inches. Dividing 80 square inches by 24.12 inches results in approximately 3.33 inches --- the polygon's apothem measures 3.32 inches.

References

• Math Open Reference: Apothem of a Regular Polygon
• Mathwords: Area of a Regular Polygon

Resources

• Calculator Soup: Regular Polygon Calculator