Octagon Area Calculator – Regular Octagon from Any Dimension
A regular octagon is an eight-sided polygon where every side is equal in length and every interior angle measures exactly 135°. The most familiar example is the stop sign, but regular octagons also appear in architecture (tower cross-sections, floor tiles, baptisteries), engineering (bolts with octagonal heads), and board game design. This calculator finds the enclosed area of a regular octagon from whichever dimension you have: side length, circumradius, or apothem. All other geometric properties are derived simultaneously.
Area Formula
The standard formula for the area of a regular octagon with side length a is:
A = 2(1 + √2) × a² ≈ 4.828427 × a²
This constant comes from the general regular-polygon area formula A = (n × a²) / (4 × tan(π/n)) evaluated at n = 8. Because tan(π/8) = √2 − 1, the expression simplifies to 2(1 + √2) × a². For a side of 5 m the area is approximately 120.71 m².
Using Other Input Dimensions
If you do not know the side length directly, the calculator accepts two alternative measurements and converts them to side length first:
- Circumradius R — the distance from the center to any vertex. Conversion:
a = 2R · sin(π/8) ≈ 0.7654 × R. - Apothem r — the perpendicular distance from the center to the midpoint of any side (also the inradius). Conversion:
a = 2r · tan(π/8) ≈ 0.8284 × r.
Derived Properties
From side length a, all other measurements follow directly:
- Perimeter:
P = 8a - Apothem (inradius):
r = a(1 + √2) / 2 ≈ 1.2071 × a - Circumradius:
R = a / (2 sin(π/8)) ≈ 1.3066 × a - Long diagonal D (vertex to opposite vertex):
D = 2R ≈ 2.6131 × a - Short diagonal d (vertex to vertex with one vertex between):
d = a√(4 + 2√2) ≈ 2.4142 × a - Interior angle: 135° (each of eight angles)
- Sum of interior angles: 1080°
Long Diagonal vs Short Diagonal
A regular octagon has two distinct diagonal lengths. The long diagonal (D) passes through the center and connects two directly opposite vertices — it equals twice the circumradius. The short diagonal (d) connects two vertices separated by exactly one vertex along the perimeter and equals a√(4 + 2√2). For a side of 5 m, D ≈ 13.07 m and d ≈ 12.07 m.
Unit Flexibility
Select any length unit for your input (mm, cm, m, km, inches, feet, or yards). You can independently choose a different unit for the area output — for example, input a side in centimeters but display the area in square meters. The calculator normalizes all values to meters internally before computing, then converts the result to your chosen area unit.
Real-World Applications
Traffic engineers and sign manufacturers reference octagon formulas when specifying stop sign blanks; the "inscribed circle" size determines the minimum clear text area. Architects designing octagonal rooms, towers, or baptisteries calculate floor areas using exactly this formula. Tile setters laying octagonal ceramic or stone patterns estimate material coverage by multiplying the unit-cell area by the number of tiles. Hardware engineers use octagon geometry when designing cap nuts, recessed bolts, and fasteners with octagonal drive heads. Game designers use regular octagon grids as an alternative to square or hex grids.
Accuracy
All calculations use IEEE 754 double-precision floating-point arithmetic, which provides approximately 15–16 significant digits. You can display results with 0 to 10 decimal places using the precision selector. The dominant source of error in practical use is the precision of your physical measurement, not the calculator's arithmetic.