All eight sides of an octagon are equal in length, and all eight angles are equal in size. This uniformity creates a direct relationship between the length of a side and the octagon area. Therefore, if you already know the area, you can derive the side length using the following formula, where "sqrt" means to take the square root: Length=sqrt(Area/(2+2*sqrt(2)))
If you know only the width of the octagon, measured perpendicularly between opposing sides, simply divide that width by 2.41 to find the side length. In the previous example, dividing the width of 7.76 inches by 2.41 correctly gives you the side length of 3.22 inches. For greater precision, use the original formula and simplify yourself: Length=Width*sqrt(2)/(2+sqrt(2))
Simplify the "2+2*sqrt(2)" part of the equation by taking the square root of 2, multiplying the result by 2 and adding 2. Therefore, this part of the equation simplifies to approximately 4.83.
Divide the area by the simplified figure in the previous step. For example, if you had an area of 50 square inches, divide 50 by 4.83 to get 10.35 square inches.
Take the square root of the results to calculate the length of one side. Continuing with the example, taking the square root of 10.35 square inches gives you a side length of 3.22 inches.
About the Author
C. Taylor embarked on a professional writing career in 2009 and frequently writes about technology, science, business, finance, martial arts and the great outdoors. He writes for both online and offline publications, including the Journal of Asian Martial Arts, Samsung, Radio Shack, Motley Fool, Chron, Synonym and more. He received a Master of Science degree in wildlife biology from Clemson University and a Bachelor of Arts in biological sciences at College of Charleston. He also holds minors in statistics, physics and visual arts.