Choose a known measurement
The calculator converts your input into an equivalent side length first.
Example octagons
Regular octagon formulas
A regular octagon has eight equal sides and eight equal interior angles. Let s be the side length, a the apothem, and R the circumradius.
P = 8s
A = 2(1 + √2)s²
a = s / [2 tan(22.5°)]
R = s / [2 sin(22.5°)]
d₁ = s√(2 + √2)
d₂ = s(1 + √2)
d₃ = s√(4 + 2√2)
Wf = s(1 + √2)
Wv = s√(4 + 2√2)
135°
45°
1080°
Using the calculator
- Select the measurement you already know.
- Enter a positive value and choose its unit.
- Select output units, decimal precision, and number formatting.
- Open a specialized mode for borders, prisms, coordinates, cutting layouts, or cost estimates.
- Press Calculate octagon.
- Review the result cards, formula substitutions, coordinate table, and diagram.
- Copy, print, or export the result as CSV, JSON, PDF, or SVG.
Understanding a regular octagon
Equal sides and equal angles
A regular octagon contains eight congruent sides. Every interior angle measures 135 degrees. Every exterior angle measures 45 degrees. The central angles are also 45 degrees because one full rotation is divided into eight equal parts.
Apothem and circumradius
The apothem runs from the center to the midpoint of a side. It is perpendicular to that side. The circumradius runs from the center to a vertex. The apothem is also the radius of the inscribed circle. The circumradius belongs to the circle passing through all eight vertices.
Three diagonal families
An octagon has twenty diagonals. Their lengths are not all equal. A short diagonal skips one vertex. A medium diagonal skips two vertices. A long diagonal joins opposite vertices. The long diagonal equals the width across corners and the circumscribed-circle diameter.
Area by triangular decomposition
Lines from the center to every vertex divide the octagon into eight congruent isosceles triangles. Each triangle has base s and height a. Their combined area is eight times one-half sa. This gives A = 4sa, which is equivalent to the common radical formula.
Construction from a square
A regular octagon can be cut from a square by removing four identical right-isosceles corner triangles. The corner marks must be placed at the same offset along both adjacent square edges. The resulting diagonal cut equals every remaining straight edge.
Miter cuts
A frame built from eight equal pieces usually uses a 22.5-degree miter on each end. Two matching miters form the 45-degree exterior turn. Real saws, stock width, kerf, joint gaps, and tolerance may require practical adjustment.
Applications
Regular octagons appear in signs, tables, windows, gazebos, floor layouts, columns, frames, tiles, machine parts, landscaping, and decorative panels. The across-flat dimension often controls fit inside an opening. The across-vertex dimension controls the minimum square envelope.
Tolerances and real materials
Exact geometry assumes perfect dimensions. Fabrication includes blade kerf, material movement, corner rounding, joint gaps, coating thickness, and measuring error. Use a sensible waste allowance. Verify critical dimensions with a drawing or physical test piece.
Regular octagon questions
How many sides does a regular octagon have?
It has eight equal sides. The prefix “octa” refers to eight.
What is each interior angle?
Each interior angle is 135 degrees. The full interior-angle sum is 1080 degrees.
How many diagonals does an octagon have?
An octagon has twenty diagonals. A regular octagon has three distinct diagonal lengths.
Is the apothem the same as the radius?
The apothem is the inradius. It reaches a side midpoint. The circumradius reaches a vertex.
What angle should an octagonal frame miter use?
Each matching frame end commonly uses a 22.5-degree miter. Together, two miters create a 45-degree turn.
How is an octagon cut from a square?
Mark equal offsets from every square corner. Connect each pair of adjacent marks. Cut along the four diagonal lines.
Can the calculator produce vertex coordinates?
Yes. Enter center coordinates, starting angle, direction, and side length. The calculator lists all eight vertices.
Can the results be exported?
Yes. The page supports CSV, JSON, coordinate CSV, SVG, PDF, printing, copying, and shareable form links.
Why can rounded values differ slightly?
Most octagon formulas contain irrational values. Decimal rounding can create tiny differences between related displayed measurements.
Which width is used for a stop sign?
Published sign sizes commonly refer to width across opposite flats. Confirm the applicable specification before fabrication.
Does the cost mode include waste?
Yes. It can add a waste percentage, labor, quantity, and tax to the selected material basis.
Can it calculate octagonal prism mass?
Yes. Add height and material density. The calculator reports estimated mass in kilograms and pounds.