Octagon Calculator: Area, Perimeter, Apothem & More
Calculate area, perimeter, apothem, circumradius, and diagonals of any regular octagon from its side length — instantly with full formulas.
Enter the side length of a regular octagon and click Calculate to get all key measurements in one step.
Octagon Calculator: Area, Perimeter, Apothem & More
Calculate area, perimeter, apothem, circumradius, and diagonals of any regular octagon from its side length — instantly with full formulas.
About the octagon calculator
A regular octagon is an eight-sided polygon in which all sides are equal in length and all interior angles are equal. Each interior angle of a regular octagon measures exactly 135°, and the sum of all interior angles is (8−2)×180° = 1080°. The octagon is one of only three regular polygons that can tile the plane when combined with squares, a fact exploited by Islamic geometric art and by floor-tile designers throughout history.
From a single side length a, all other measurements of a regular octagon can be derived analytically using exact algebraic expressions involving √2. The area is 2(1+√2)a², which for a = 1 gives approximately 4.8284 square units. The perimeter is simply 8a. The apothem (also called the inradius) is the distance from the centre to the midpoint of any side, equal to a(1+√2)/2. The circumradius is the distance from the centre to any vertex, equal to (a/2)√(4+2√2).
The octagon has three distinct classes of diagonal. The short diagonal connects two vertices separated by one intermediate vertex (spanning two sides), with length a√(2+√2) ≈ 1.848a. The medium diagonal connects vertices separated by two intermediate vertices (spanning three sides), with length a(1+√2) ≈ 2.414a. The long diagonal (diameter) connects directly opposite vertices and has length a√(4+2√2) ≈ 2.613a, which equals exactly 2R (twice the circumradius). This calculator reports all three diagonal types.
Octagons appear throughout architecture, engineering, and everyday objects. Stop signs are regular octagons, standardised internationally because their shape is unmistakeable even from a distance and when partially obscured. The Baptistery of Florence, the dome base of Saint Basil's Cathedral, and the courtyard of the Castel del Monte are famous octagonal structures. In engineering, octagonal cross-sections balance structural efficiency with ease of fabrication: they approximate a circle more closely than a hexagon while still being easy to measure and cut.
In tile and paving design, the octagon-plus-square pattern requires tiles in a 1:(1+√2)² area ratio. In woodworking and metalworking, turners and machinists use octagonal stock as an intermediate step when rounding cylindrical parts. Knowing the relationship between a regular octagon's side and its circumradius is essential for laying out octagonal gazebo footings, cutting octagonal mirror frames, or designing octagonal column bases.
This calculator uses exact IEEE-754 double-precision arithmetic for all computations, providing results accurate to ten significant digits for any positive side length. All seven output values — area, perimeter, apothem, circumradius, short diagonal, medium diagonal, and long diagonal — are computed from a single side-length input.
Octagon examples
Four examples from common applications of regular octagon calculations.
| Side Length (a) | Area | Context |
|---|---|---|
| a = 10 | Area ≈ 482.843 | A standard octagon with side 10 units. Perimeter = 80, apothem ≈ 12.071, circumradius ≈ 13.066. Useful for calculating a gazebo floor area. |
| a = 2.5 | Area ≈ 30.178 | Small octagon, e.g. a logo design element. Perimeter = 20, apothem ≈ 3.018. |
| a = 120 | Area ≈ 69,529 | Large architectural octagon with side 120 cm (representing a gazebo base). Perimeter = 960 cm, circumradius ≈ 156.8 cm. |
| a = 7.75 | Area ≈ 289.77 | Fractional side length to verify precision. Demonstrates the calculator handles non-integer inputs correctly. |
How to use the octagon calculator
- Enter the side length of the regular octagon in the Side Length (a) field. Use any unit — cm, metres, inches, feet — and the results will be in the same unit (and unit² for area).
- Click Calculate. The results panel shows area, perimeter, apothem, circumradius, short diagonal, and long diagonal simultaneously.
- Click Reset to clear the input and start with a new side length.
- Use the example buttons below the worked-examples table to instantly load common side lengths — 10, 2.5, or 120 — into the calculator.
- To reverse-engineer the side length from a known area, solve a = √(Area / (2(1+√2))) using the formula shown below the results.
Octagon calculator FAQ
What is the formula for the area of a regular octagon?
The area of a regular octagon with side length a is A = 2(1+√2)a². For a = 10, that gives 2(1+1.41421)×100 = 2×2.41421×100 ≈ 482.84 square units. This formula comes from dividing the octagon into a central rectangle, four rectangles along the sides, and four right isosceles triangles at the corners, then summing their areas.
What is the apothem of an octagon?
The apothem (inradius) is the perpendicular distance from the centre of the octagon to the midpoint of any of its sides. For side length a it equals a(1+√2)/2. The apothem is the radius of the largest circle that fits inside the octagon. It is used in the alternative area formula A = apothem × perimeter / 2, which applies to any regular polygon.
What is the circumradius of a regular octagon?
The circumradius is the distance from the centre to any of the eight vertices. For side length a it equals (a/2)√(4+2√2). The circumradius is the radius of the smallest circle that circumscribes the octagon. It is always larger than the apothem; for an octagon the ratio circumradius/apothem = √(4+2√2) / (1+√2) ≈ 1.0824.
How many diagonals does a regular octagon have?
A regular octagon has 8(8−3)/2 = 20 diagonals in total. These fall into three length classes: 8 short diagonals of length a√(2+√2) ≈ 1.848a (connecting vertices 2 apart), 8 medium diagonals of length a(1+√2) ≈ 2.414a (connecting vertices 3 apart), and 4 long diagonals (diameters) of length a√(4+2√2) ≈ 2.613a connecting opposite vertices. This calculator reports all three types.
Why is a stop sign octagonal?
Stop signs are regular octagons because the shape is immediately distinctive — it is the only common road sign with eight sides — and because it is symmetric enough to be readable from any approach angle. The octagonal shape is specified by the Vienna Convention on Road Signs, making it a near-universal standard. Even with part of the sign obscured by snow or a vehicle, the distinctive red octagon is unmistakeable.
Can I calculate the octagon from area or perimeter instead of side length?
Yes, by inverting the formulas. If you know the perimeter P, side = P/8. If you know the area A, side = √(A / (2(1+√2))). If you know the apothem r, side = 2r/(1+√2). Once you have the side length, enter it into this calculator to get all other measurements instantly.