How this instrument works
A regular octagon has eight equal sides and eight interior angles of exactly 135°, and its area formula falls out of the same construction that works for any regular polygon: draw a line from the centre to each vertex and the shape splits into eight congruent isosceles triangles. A full turn is 360°, so each triangle's angle at the centre is exactly 45°; bisecting that angle to solve for the apothem is where the tan(π⁄8) in A = 8s² ⁄ (4·tan(π⁄8)) comes from. Set n = 8 in the general regular-polygon formula A = ns² ⁄ (4·tan(π⁄n)) and this is exactly what falls out.
Unlike most regular polygons, the octagon's tangent term collapses to a clean number: tan(π⁄8), which is tan(22.5°), equals √2 − 1 exactly, a result of the half-angle identity applied to the 45° angle above. Substituting that value and simplifying clears every trig function from the formula, leaving A = 2(1 + √2)s². The constant 1 + √2 ≈ 2.414214 is called the silver ratio — the octagon's own counterpart to the golden ratio that turns up in the pentagon and decagon, produced by the identical trick of bisecting a regular polygon's own central angle, but a distinct constant with its own continued fraction of nothing but 2s.
That closed form hides a carpenter's shortcut. Start with a square of side L and saw an equal right-triangle corner off each of the four corners, cutting a length x = L(1 − 1⁄√2), about 29.29% of L, from every corner; the eight new edges come out equal, forming a true regular octagon with side s = L(√2 − 1). It is the method behind gazebo decks, picture frames, and stop-sign blanks alike. A single regular octagon cannot tile a flat surface on its own, since 360° never divides evenly by its 135° corner, but pair it with a square at every gap — 135° + 135° + 90° closes the full 360° exactly — and the two shapes interlock in the truncated square tiling visible in countless sidewalk and tiled-floor patterns.
- Enter your octagon's edge length into the Side length field — any unit works, since Area (regular octagon) and Perimeter come back in that same unit, area squared.
- Read Area (regular octagon) for the enclosed surface, computed as A = 8s² ⁄ (4·tan(π⁄8)).
- Read Perimeter for the total boundary length, simply eight times the Side length.
- Building one from a square blank? Cut about 29.29% of the square's side off each corner and check the result against the worked example below.
- To sanity-check the sheet, double the Side length and confirm Area rises fourfold rather than twofold, since area follows the square of a length.
Worked example — a gazebo octagon with 4-metre sides
A garden gazebo floor is framed as a regular octagon with every side cut to exactly 4 metres, so s = 4. Area (regular octagon) comes out to 8 × 4² ⁄ (4 × tan(π⁄8)) = 77.25483399593904 square metres, and Perimeter reads 8 × 4 = 32 metres — the length of sill plate needed to ring the frame.
The closed form checks the same number a second way, with no tangent involved: 2 × (1 + √2) × 4² = 2 × 2.414213562373095 × 16 = 77.25483399593904, the identical figure, digit for digit. Building this octagon by cutting the corners off a square lands on the same shape: start with a square measuring 9.65685424949238 m per side, saw 2.82842712474619 m off each corner — 29.29% of the square's side — and the eight remaining edges settle at exactly 4 m apiece, matching Perimeter's 32 m total exactly.
Questions
What is the formula for the area of a regular octagon?
A = 8s² ⁄ (4·tan(π⁄8)), where s is one side — the general regular-polygon formula A = ns² ⁄ (4·tan(π⁄n)) with n = 8. Because tan(π⁄8) has the exact value √2 − 1, the formula also simplifies to A = 2(1 + √2)s²; with s = 4 both routes return 77.25483399593904, this sheet's own reference value.
Why does tan(π⁄8) simplify to a whole expression instead of an endless decimal?
Because π⁄8 is 22.5°, half of the 45° angle each of the eight triangles makes at the octagon's centre, and the half-angle identity for 45° collapses cleanly: tan(22.5°) = √2 − 1 exactly, with nothing left over. Substituting that value is what turns A = 8s² ⁄ (4·tan(π⁄8)) into the trig-free closed form A = 2(1 + √2)s².
What is the 'silver ratio' and how does it relate to an octagon?
It is the constant 1 + √2 ≈ 2.414214 left behind once the octagon's area formula is stripped of its tangent, in A = 2(1 + √2)s². It plays the same role for the octagon that the golden ratio plays for the pentagon and decagon — a single irrational constant produced by bisecting the shape's own central angle — but it is a distinct number, with its own continued fraction of nothing but 2s rather than 1s.
How do I lay out a regular octagon by cutting the corners off a square?
Saw a right-triangle corner off each of the square's four corners, removing a length of 1 − 1⁄√2 (about 29.29%) of the square's side from every one. A 9.65685424949238 m square trimmed this way, 2.82842712474619 m off each corner, produces a regular octagon with 4 m sides — the same shape and numbers as this sheet's worked example.
Can regular octagons tile a flat surface on their own?
No — an octagon's interior angle is 135°, and 360° does not divide evenly by 135°, so copies meeting at a point always leave a gap or force an overlap. Pair each gap with a square instead: 135° + 135° + 90° closes the full 360° exactly, which is why octagons and squares interlock in the truncated square tiling seen in countless sidewalk and tiled-floor patterns.
What's a common mistake when measuring a real octagon for this calculator?
Entering the flat-to-flat width instead of one edge length. Stop-sign blanks and octagonal fixtures are often specified by that across-the-flats distance, which is twice the apothem, not a side — feeding it straight into Side length will overstate both Area (regular octagon) and Perimeter.