Introduction
An octagon is a shape with eight sides and eight angles. You see octagons every day. The most common example is a stop sign. This octagon calculator helps you find important measurements like side length, area, perimeter, and diagonals. Just enter one known value, and the calculator does the rest of the math for you. It works for regular octagons, which means all eight sides are the same length and all eight angles are equal. Each interior angle of a regular octagon measures 135 degrees. This tool gives fast, accurate results for homework problems, building projects, or plain curiosity about this eight-sided shape.
How to Use Our Octagon Calculator
Enter the side length of your regular octagon, and this calculator will instantly find the area, perimeter, diagonal length, and other important measurements for you.
Side Length: Type in the length of one side of your octagon. Since a regular octagon has eight sides that are all the same length, you only need to enter one side. Use any unit you like, such as inches, centimeters, meters, or feet. The results will be in the same unit you choose.
What Is a Regular Octagon?
A regular octagon is a flat shape with 8 equal sides and 8 equal angles. You see octagons in everyday life. The most common example is a stop sign. Because all the sides and angles are the same, a regular octagon has a lot of symmetry, which makes it easy to calculate its properties once you know just one measurement.
Key Properties of a Regular Octagon
Every regular octagon shares certain fixed angle measurements. Each interior angle is always 135°, and each exterior angle is always 45°. The sum of all interior angles in an octagon is 1,080°. A regular octagon also has exactly 20 diagonals, which are lines that connect non-adjacent vertices inside the shape. You can verify the number of diagonals using the combination calculator, since diagonal count is based on choosing 2 vertices from 8 and subtracting the 8 sides.
Understanding the Different Measurements
A regular octagon has several important measurements, all of which can be figured out from the side length (a) alone:
- Perimeter (P): The total distance around the octagon. Since there are 8 equal sides, the perimeter is simply P = 8a.
- Area (A): The space inside the octagon. The formula is A = 2(1 + √2) × a², which is roughly 4.828 × a².
- Short Diagonal (f): A line connecting two vertices that have one vertex between them. Its length is a × √(2 + √2), or about 1.848 × a.
- Medium Diagonal (e): A line connecting two vertices with two vertices between them. It equals a × (1 + √2), or about 2.414 × a. This is also the height of the octagon measured across a flat side.
- Long Diagonal (d): The longest diagonal, which passes straight through the center and connects opposite vertices. Its length is a × √(4 + 2√2), or about 2.613 × a.
- Circumradius (R): The radius of the circle that passes through all 8 vertices (the circumscribed circle). It equals half the long diagonal.
- Inradius or Apothem (r): The radius of the circle that fits perfectly inside the octagon, touching the midpoint of each side (the inscribed circle). Its formula is r = a(1 + √2) / 2, or about 1.207 × a.
How This Calculator Works
This octagon calculator lets you enter any single known property (side length, perimeter, area, any diagonal, circumradius, or inradius), and it computes all the other properties for you. Because every measurement in a regular octagon is directly tied to the side length through a fixed ratio, knowing just one value is enough to determine everything else. Simply type a number into any field, and the calculator does the rest. The area formula involves a square root, and you may want to manage precision in your results.
Where Regular Octagons Appear
Beyond stop signs, regular octagons show up in architecture, tile patterns, flooring designs, umbrella frames, and fortress layouts. Architects and engineers use octagonal shapes because they offer a good balance between a circle and a square: they enclose a large area relative to their perimeter while still having flat sides that are easy to build. Understanding octagon geometry is valuable for planning construction projects, cutting materials, or solving math problems in school.