A regular octagon has eight equal sides and eight 135° angles. You see it on every stop sign, in gazebos, umbrellas, tiles, and window frames. It can be pictured as a square with its four corners clipped off, which is the key to most of its formulas: the factor 1 + √2 ≈ 2.414 appears again and again. Enter any one measurement and this calculator returns the rest, including the three different diagonal lengths and the corner cut needed to make one from a square.
How to use the octagon calculator
- Choose the measurement you know: side, width across flats, width across corners, perimeter or area.
- Enter the value and choose the unit.
- Read the area on the tape (or the side if you entered the area), followed by all diagonals, the circumradius, the apothem and the corner-cut length. Exact forms such as 200 + 200√2 appear when the side or perimeter is a simple number.
Octagon formulas
| Measurement | Formula in terms of side s | Approximate |
|---|---|---|
| Width across flats (middle diagonal) | (1 + √2)s | 2.4142s |
| Width across corners (long diagonal) | √(4 + 2√2) s | 2.6131s |
| Short diagonal | √(2 + √2) s | 1.8478s |
| Perimeter | 8s | |
| Area | 2(1 + √2)s2 | 4.8284s2 |
| Apothem | (1 + √2)s/2 | 1.2071s |
| Circumradius | √(4 + 2√2) s/2 | 1.3066s |
| Corner cut from a square | s/√2 | 0.7071s |
The area formula follows from the clipped-square picture. A square of side F = (1 + √2)s loses four right isosceles triangles with legs s/√2, each of area s2/4:
Worked examples
Side 10 in. Across flats 10 + 10√2 ≈ 24.14 in; across corners ≈ 26.13 in; short diagonal ≈ 18.48 in; perimeter 80 in; area 200 + 200√2 ≈ 482.84 in².
Check by clipping a square: a 24.142-inch square has area 582.84 in². Removing four corner triangles with 7.071-inch legs (½ × 7.071² = 25 in² each) leaves 582.84 − 100 = 482.84 in² ✓.
A stop sign. The standard US stop sign is 30 inches across flats. Its side is 30 ÷ (1 + √2) ≈ 12.43 in, and its face area is about 745.6 in², a little over 5 square feet.
Building an octagon from a square
Woodworkers and quilters often make octagons by trimming squares. If the square’s side is W:
- The finished octagon’s side is s = W/(1 + √2) ≈ 0.4142W.
- Measure c = W/(2 + √2) ≈ 0.2929W from each corner along both edges.
- Connect the marks and cut off the four triangles.
For an 18-inch square, mark about 5.27 inches from each corner; the resulting octagon has 7.46-inch sides.
For an octagonal frame or gazebo built from eight boards, each board’s ends are cut at 22.5°, half of the 45° exterior angle, so that eight joints close the shape.
Angles and symmetry
Each interior angle is 135°, the eight angles total 1,080°, and each exterior angle is 45°. The octagon has eight lines of symmetry and repeats every 45° of rotation. Regular octagons cannot tile a floor on their own; the classic octagon-and-square tile pattern fills the gaps with small squares whose sides equal the octagon’s.
Related tools
The hexagon calculator handles six-sided shapes, and the regular polygon calculator any number of sides. To check the corner triangles, use the Pythagorean theorem calculator, and for the starting square, the square calculator.
Frequently asked questions
What is the area of a regular octagon?
A = 2(1 + √2)s² ≈ 4.828s², where s is the side length. An octagon with 10-inch sides has area 200 + 200√2 ≈ 482.84 square inches.
How do I find the side from the width?
If the width across flats (between parallel sides) is F, the side is s = F/(1 + √2) ≈ 0.4142F. A stop sign that is 30 inches across flats has sides of about 12.43 inches.
How many diagonals does an octagon have, and how long are they?
Twenty, in three lengths: short ones skipping one corner, √(2 + √2)·s ≈ 1.848s; medium ones equal to the width across flats, (1 + √2)s ≈ 2.414s; and long ones through the center, √(4 + 2√2)·s ≈ 2.613s.
How do I cut an octagon from a square board?
Mark a distance c = s/√2 ≈ 0.2929 × the square's width from each corner along both edges, then cut off the four corner triangles. The remaining sides are all equal.