The trigonometry of a staircase
A stair is a stack of identical right triangles. Once you see that, every stair measurement is one of three functions.
The one idea
Every stair question reduces to a right triangle. The rise is the opposite side, the going is the adjacent side, and the line joining the nosings — the pitch line — is the hypotenuse.
That triangle repeats, identically, once per step. The whole flight is the same triangle scaled up: total rise opposite, total run adjacent, stringer as the hypotenuse. Solve the small one and you have the big one, and vice versa.
Why this matters practically: it means you never need to measure a stair's angle or its stringer. Both fall out of two numbers you already have.
1. The angle — tangent
Tangent relates the two sides you know to the angle you want:
tan θ = rise / going
θ = arctan(rise / going)
With a 150mm rise and a 250mm going: tan θ = 150/250 = 0.6, so
θ = arctan(0.6) = 31°. Comfortably inside the 30–37° band.
The same works at flight scale — arctan(total rise / total run) gives the identical
angle, because the triangles are similar. If your two answers differ, one of your measurements is
wrong, which makes this a useful check.
2. The stringer — Pythagoras
stringer² = total rise² + total run²
stringer = √(total rise² + total run²)For the 15-riser stair in the stair calculator's example — 108″ of rise over 145.6″ of run — the pitch line measures 181.3″. Order stock longer than that: the figure is the pitch line, not the board, and the plumb cuts at each end need material beyond it.
3. The floor opening — tangent, rearranged
This is the one people find hardest, and it is the same function read backwards. You know the vertical drop you need (headroom plus the thickness of the floor above) and the angle; you want the horizontal distance.
tan θ = drop / run
run = drop / tan θThe floor opening calculator does exactly this.
4. Where sine and cosine come in
Less often, but they do. Measuring along the rake rather than horizontally — a handrail, a wall panel, a carpet runner — means working from the hypotenuse:
sin θ = rise / stringer → rise = stringer × sin θ
cos θ = going / stringer → going = stringer × cos θ
A handrail 3600mm long on a 35° stair rises 3600 × sin 35° = 2065mm and covers
3600 × cos 35° = 2949mm horizontally.
5. Winders — radians, not degrees
Tapered treads are the one place the arithmetic leaves right triangles. The going across a winder is an arc length, and arc length wants radians:
arc = radius × angle in radians
radians = degrees × π / 180
A 90° turn across three winders is 30° each. On the walk line 12″ out from the newel, each
winder's going is 12 × (30 × π/180) = 6.28″ — well under the 10″ the IRC
requires at the walk line, which is precisely why three winders across a tight corner so often
fails and why the walk line has to be further out than people expect.
A worked example, end to end
Floor to floor 2745mm. Target riser 180mm.
- Risers:
round(2745 / 180) = 15 - Riser height:
2745 / 15 = 183mm - Going from Blondel:
630 − 2(183) = 264mm - Treads:
15 − 1 = 14 - Total run:
14 × 264 = 3696mm - Angle:
arctan(183 / 264) = 34.7° - Stringer:
√(2745² + 3696²) = 4604mm
Seven lines, three functions, and the stair is fully specified.
The check worth doing
Compute the angle twice — once from one step, once from the whole flight. They must agree. If they do not, either your total rise or your riser count is wrong, and you have caught it before cutting rather than after.