Every formula used across the site, in the open, with the reasoning behind each. Nothing here is a black box.
Stair geometry starts from the total rise — the finished floor-to-floor height. Dividing it by a target riser height and rounding to a whole number gives the step count, because you cannot have a fractional step. Dividing the total rise back by that whole count gives the actual, uniform riser height.
There is one fewer tread than riser because the topmost riser lands on the floor above, which serves as the last tread.
The stringer is the sloped board the steps are cut from. Its length is the hypotenuse of the triangle formed by total rise and total run.
Comfortable domestic stairs sit around 30–38°. Codes cap the maximum: UK Part K, for example, limits private stairs to 42°.
Named for Nicolas-François Blondel, this 17th-century relationship still governs stair comfort. It reflects the natural human stride: as risers get taller, treads should get shorter, and vice versa.
A 7-inch riser with an 11-inch tread gives 2×7 + 11 = 25 — the textbook ideal.
Winder treads are pie-shaped, narrow at the inside of the turn and wide at the outside. Because each tread is an arc, its depth at any radius equals that radius times the angle the tread sweeps (in radians).
Codes set two minimums: a depth at the narrow end (6.0″ under IRC) and a depth along the walking line, measured a fixed offset from the narrow side (10.0″ at 12.0″ from the narrow edge under IRC).
A spiral stair wraps treads around a center column. Clear width is the outer radius minus the column radius. Each tread is an arc, so the usable tread depth is measured along a walking line near the outer edge. The handrail follows a helix, so its length combines the outer arc with the total rise.
Compliance values are verified against primary sources per jurisdiction: IRC, IBC, UK Part K, and Australian NCC. Eurocode values are representative estimates, since EN has no single prescriptive stair table.