Octagon Calculator

What is a Regular Octagon?

A regular octagon is a flat geometric shape with eight equal sides and eight equal interior angles. Each interior angle of a regular octagon is exactly $135^\circ$ ($2.35619$ radians), and the sum of all interior angles is $1080^\circ$. Regular octagons are common in engineering, architectural design (like stop signs), and decorative tiling.

Important Formulas for Regular Octagons

All properties of a regular octagon can be determined from a single dimension, such as the side length $s$. Below are the key geometric formulas:

• Perimeter ($P$): The total distance around the octagon is: $$P = 8s$$
• Area ($A$): The surface area enclosed is given by: $$A = 2(1 + \sqrt{2})s^2 \approx 4.828427 s^2$$
• Apothem / Inradius ($a$): The distance from the center to the midpoint of any side (radius of the inscribed circle): $$a = \frac{1 + \sqrt{2}}{2}s \approx 1.207107 s$$
• Circumradius ($R$): The distance from the center to any vertex (radius of the circumscribed circle): $$R = \frac{\sqrt{4 + 2\sqrt{2}}}{2}s \approx 1.306563 s$$
• Long Diagonal ($d_1$): The maximum distance between opposite vertices: $$d_1 = 2R = \sqrt{4 + 2\sqrt{2}}s \approx 2.613126 s$$
• Short Diagonal ($d_2$): The distance between parallel sides (equivalent to twice the apothem): $$d_2 = 2a = (1 + \sqrt{2})s \approx 2.414214 s$$

Real-World Applications

From stop signs to polygon meshes in game design, octagons are ideal for transitioning between square layouts and circular layouts. The ratio of the area of an octagon to the bounding square is approximately $82.8\%$, making it an efficient shape for structural design and material optimization.