Octagon Calculator
Enter the side length of a regular octagon to find its area, perimeter and width.
A regular octagon has eight equal sides. Its perimeter is 8 times the side length. Its area is 2 ร (1 + โ2) ร sideยฒ, about 4.828 ร sideยฒ. The width across the flat sides is (1 + โ2) ร side, roughly 2.414 times the side length.
How to Use This Calculator
Enter the length of a single side of a regular octagon โ one where all eight sides and angles are equal. The calculator returns the area, the perimeter, and the width measured across the flat sides (the distance a wrench would span on an eight-sided nut, for example).
Enter the side in any unit; area comes back in that unit squared, and perimeter and width in that unit.
How the Calculation Works
The perimeter is simply eight side lengths. The area formula, 2 ร (1 + โ2) ร sideยฒ, comes from splitting the octagon into triangles and rectangles and summing them; the constant 2(1 + โ2) works out to about 4.828.
The flat-to-flat width โ the span across opposite parallel sides โ is (1 + โ2) times the side, about 2.414 times, because a side plus two 45-degree diagonal projections make up that span.
area = 2(1 + โ2)ยทsยฒ ยท perimeter = 8s ยท width = (1 + โ2)ยทs
Worked Example
For a regular octagon with a side of 5: the perimeter is 8 ร 5 = 40. The area is 2 ร (1 + โ2) ร 5ยฒ = 4.828 ร 25 = 120.71.
The flat-to-flat width is (1 + โ2) ร 5 = 2.414 ร 5 = 12.07.
Assumptions and Limitations
- Applies to regular octagons only โ all sides and angles equal
- Area is in square units; perimeter and width in the same unit as the side
- Width given is flat-to-flat, not corner-to-corner
- Results are rounded for display
Who Uses an Octagon Calculator
Woodworkers, builders, and students use this to find the area, perimeter, and width of a regular octagon from its side length. It is handy for building a gazebo, table, planter, or stop-sign-shaped project, and for geometry homework. Enter one side to get every key measurement of the shape.
Related Calculators
Perimeter Calculator
Find the perimeter of a rectangle, square, triangle or circle.
Open calculator โDiagonal Calculator
Find the diagonal of a rectangle, square or box.
Open calculator โSpecial Right Triangles Calculator
Solve 45-45-90 and 30-60-90 triangles from one side.
Open calculator โCentroid Calculator (Triangle)
Find a triangle's centroid from its three vertices.
Open calculator โFrequently Asked Questions
What is the area of a regular octagon?
It is 2 ร (1 + โ2) ร sideยฒ, about 4.828 times the side squared. A regular octagon with a 5-unit side has an area of about 120.71 square units.
What is the perimeter of an octagon?
For a regular octagon it is eight times the side length, since all eight sides are equal. A side of 5 gives a perimeter of 40.
What is the flat-to-flat width of an octagon?
It is (1 + โ2) times the side, about 2.414 ร side โ the distance across opposite parallel sides. For a 5-unit side that is about 12.07 units.
Does this work for an irregular octagon?
No. These formulas assume a regular octagon with equal sides and angles. An irregular eight-sided shape must be split into triangles and measured piece by piece.
Why is the octagon area constant about 4.828?
Because 2 ร (1 + โ2) equals 4.8284. It is a fixed geometric constant for regular octagons, just as a circle's area uses the fixed constant ฯ.
Looking for a different calculator?
CalculatorPlus has free tools across finance, construction, math, health and more โ each one showing the formula and a worked example.