Stair formula calculator

Every stair starts with total rise, the vertical distance from finished floor to finished floor. You pick a target riser height, then let the math round it to a whole number of steps: risers = round(total rise ÷ target riser). You can't build a fraction of a step, so this rounding step is not optional.

Once you know the riser count, go back and divide the total rise by it to get the real riser height: riser height = total rise ÷ risers. That number is what actually gets cut, and it will rarely match your original target exactly.

A flush-top stair (top step level with the upper floor) always has one fewer tread than riser, because the top floor itself acts as the last step: treads = risers − 1. If you're building a platform-style bottom step or an open-top design, that count can shift, so confirm which convention your plan uses before ordering lumber.

Multiply the tread count by your chosen tread depth to get the total horizontal run of the stair: total run = treads × tread depth. This is the number you check against the floor space you actually have before you cut anything.

The stair angle is the pitch of the stringer line above horizontal, and it comes straight out of basic trigonometry: angle = atan(riser height ÷ tread depth). Most residential stairs land between about 30 and 37 degrees; steeper stairs feel more like a ladder, shallower stairs eat up more floor space.

The same right triangle also gives you slope as a percentage (rise ÷ run × 100) or as a ratio (rise : run) if your local code or drawings call for that instead of degrees.

Stringer length is the hypotenuse of the same right triangle, using the full rise and run of the flight, not a single step: stringer length = √(rise² + run²). This is the theoretical layout line down the top of the stringer board, before you add material for the bottom cut or a horseshoe-style tail.

For a 12-foot stringer board, remember you're buying length along that diagonal, not along the floor, so a shallow stair with a long run needs a noticeably longer board than a steep one with the same total rise.

Blondel's formula (2R + T) is the oldest and still the most-quoted comfort check for a stair, and it aims for a stride length that matches a natural walking pace: 2 × riser height + tread depth should land somewhere around 24 to 25 in.

Two simpler backup checks: R + T (riser plus tread) should be roughly 17 to 18 in, and R × T (riser times tread) should land around 70 to 75 sq in. None of these are code minimums, they're comfort guidelines, and a stair can pass code while failing all three if it's unusually steep or shallow.

Headroom is the clear vertical distance from the nosing line (an imaginary line touching the front edge of every tread) up to the underside of the ceiling or framing above. It isn't a formula you solve for directly on a finished stair; it's measured on site or checked against a planned floor opening, since it depends on where the opening starts and ends, not just the stair's own geometry.

The notch depth on a cut stringer is simply the riser height minus the tread board's thickness, so the first tread sits level with the rest once it's fastened on. The throat is the narrowest remaining width of stringer material below the bottom of the notches; framing guidance commonly calls for keeping at least 3½ in to 5 in of solid wood there so the stringer doesn't crack under load.

Here's every formula from this page side by side, so you can scan for the one you need without re-reading each section.