Octagon Calculator
Calculate octagon instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods. Enter your values for instant results.
Formula
Area = 2(1 + sqrt(2)) x s^2
Where s is the side length of the regular octagon. The constant 2(1 + sqrt(2)) is approximately 4.8284. This formula derives from decomposing the octagon into triangles from the center or from a square minus corner triangles.
Worked Examples
Example 1: Stop Sign Dimensions
Problem: A standard US stop sign has sides of 12.5 inches. Calculate its area, perimeter, and the distance across the sign (long diagonal).
Solution: Side length s = 12.5 inches\nArea = 2(1 + sqrt(2)) x 12.5^2 = 2 x 2.4142 x 156.25 = 754.44 sq inches\nPerimeter = 8 x 12.5 = 100 inches\nCircumradius = 12.5 / (2 x sin(pi/8)) = 12.5 / 0.7654 = 16.332 inches\nLong diagonal = 2 x 16.332 = 32.664 inches\nApothem = 12.5 x (1 + 1.4142) / 2 = 15.089 inches
Result: Area: 754.44 sq in | Perimeter: 100 in | Span: 32.66 in
Example 2: Octagonal Gazebo Foundation
Problem: An octagonal gazebo has sides of 1.5 meters. Calculate the floor area and the distance from center to each corner post.
Solution: Side length s = 1.5 m\nArea = 2(1 + sqrt(2)) x 1.5^2 = 4.8284 x 2.25 = 10.864 sq meters\nCircumradius = 1.5 / (2 x sin(pi/8)) = 1.5 / 0.7654 = 1.960 m\nApothem = 1.5 x (1 + 1.4142) / 2 = 1.811 m\nPerimeter = 8 x 1.5 = 12.0 m
Result: Floor Area: 10.86 sq m | Center to corner: 1.96 m | Perimeter: 12.0 m
Frequently Asked Questions
What is a regular octagon and what are its properties?
A regular octagon is an eight-sided polygon with all sides equal in length and all interior angles equal. Each interior angle measures exactly 135 degrees, and the sum of all interior angles is 1,080 degrees. A regular octagon has 20 diagonals and exhibits 8 lines of symmetry. It has rotational symmetry of order 8, meaning it looks identical after rotation by 45 degrees. The octagon is one of the most common polygons in everyday life, appearing in stop signs, architectural designs, decorative tiles, and MMA fighting rings. Its near-circular shape makes it an efficient use of space while maintaining flat edges that are easier to construct than curves.
How do you calculate the area of a regular octagon?
The area of a regular octagon with side length s is calculated using the formula: Area = 2(1 + sqrt(2)) x s squared. This formula derives from dividing the octagon into eight identical isosceles triangles meeting at the center, or alternatively by starting with a square and accounting for the corner cuts. The constant 2(1 + sqrt(2)) equals approximately 4.8284. You can also calculate the area using the apothem: Area = (1/2) x perimeter x apothem = (1/2) x 8s x a, where a is the perpendicular distance from center to side midpoint. For a regular octagon with side length 10, the area would be 2(1 + 1.4142) x 100 = 482.84 square units. This formula only applies to regular octagons where all eight sides are equal.
How many diagonals does an octagon have and what types exist?
A regular octagon has exactly 20 diagonals, calculated using the formula n(n-3)/2 where n equals 8. These diagonals come in three distinct types based on how many vertices they skip. Short diagonals connect vertices separated by one vertex (there are 8 of these), with length s(1 + sqrt(2)). Medium diagonals connect vertices separated by two vertices (there are 8 of these), with length s times sqrt(4 + 2sqrt(2)). Long diagonals connect opposite vertices directly through the center (there are 4 of these), with length equal to twice the circumradius. The long diagonals are the only ones that pass through the center point and they divide the octagon into congruent sections. Understanding diagonal lengths is important for structural bracing calculations and decorative pattern design.
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Is Octagon Calculator free to use?
Yes, completely free with no sign-up required. All calculators on NovaCalculator are free to use without registration, subscription, or payment.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.