The Octagon Calculator calculates the area, perimeter and overall widths of a regular (equal-sided) octagon.
From the side length, the area is A = 2(1 + √2) × s² ≈ 4.8284 × s² and the width across flats is s × (1 + √2). If you know the overall width instead — for an octagonal table, gazebo, deck or window — the second mode finds the side length for you.
Octagon Calculator
Regular octagon area, perimeter, width across flats and diagonal — from side length or width across flats.
Formula
- Area: A = 2(1 + √2) × s² ≈ 4.8284s²
- Perimeter: P = 8s
- Width across flats: W = s(1 + √2) ≈ 2.4142s
- Width across corners: D = s√(4 + 2√2) ≈ 2.6131s
- Side from width: s = W ÷ (1 + √2)
Where s = side length, W = distance between two opposite flat sides, D = distance between two opposite corners.
How the Calculation Works
Step 1: Picture the octagon as a W × W square with its four corners cut off as right triangles.
Step 2: Each cut-off triangle has legs of s ÷ √2, so W = s + 2 × s ÷ √2 = s(1 + √2).
Step 3: The area is the square minus the four triangles: W² − s², which simplifies to 2(1 + √2)s².
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| s | Side length | in, ft, cm, m |
| W | Width across flats | in, ft, cm, m |
| D | Width across corners (longest diagonal) | in, ft, cm, m |
| A | Area | in², ft², cm², m² |
Example Calculation
From the side length
Given: side = 5 cm
Result: A = 4.8284 × 25 ≈ 120.71 cm²; P = 40 cm; W ≈ 12.07 cm; D ≈ 13.07 cm
Octagonal table 48 in across
Given: width across flats = 48 in
Side: s = 48 ÷ 2.4142 ≈ 19.88 in
Result: A ≈ 1,908.7 in² ≈ 13.25 ft²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 ft² | 144 in² |
| 1 m² | 10.7639 ft² |
| 1 m² | 10,000 cm² |
Shape Properties
A regular octagon has 8 equal sides, 8 interior angles of 135° and 20 diagonals. When building one from boards, each joint is cut at 22.5° on both pieces, so that the two cuts together make the 45° turn at each corner.
Irregular octagons
If the sides are not equal, use the Polygon Area Calculator with the corner coordinates instead.
Dimensions Explained
Width across flats is measured between two parallel sides. Width across corners is measured between two opposite points and is always larger. Make sure you know which one you have before using the width mode.
Frequently Asked Questions
How do you find the area of an octagon?
For a regular octagon, A = 2(1 + √2) × side² ≈ 4.8284 × side².
How do you find the side length of an octagon from its width?
Divide the width across flats by 2.4142 (1 + √2).
What angle do you cut for an octagon?
22.5° on each end of every board, giving 135° interior corners.
What is the interior angle of an octagon?
135°, and all eight angles add up to 1,080°.
How do you find the area of an octagon from its width?
A = 2(√2 − 1) × W² ≈ 0.8284 × W².
