Every stair reduces to rise and run. Rise is the vertical climb; run is the horizontal travel. The ratio between them sets the pitch, and the pitch sets how the stair feels to climb. A tall rise with a short run is steep and tiring; a short rise with a long run is shallow and space-hungry.
Unit rise is one step's height; unit run is one tread's depth. Total rise and total run are the whole stair's vertical and horizontal spans. You design from the total rise (a fixed floor-to-floor height) and a target unit rise, which gives the step count.
The same slope can be written several ways: as an angle (arctan of rise over run), a percent grade (rise over run times 100), a ratio, or as "1 in N". Stairs are usually discussed as an angle; ramps and grading as a percent or ratio.
The Blondel rule ties rise and run together for comfort: 2×rise + run should land near 24 to 25 inches. The rise over run calculator converts between all the slope forms instantly.
Take a 108-inch total rise (a typical 9-foot floor-to-floor) with a 120-inch available run. Choosing a 7.5-inch target riser gives 108 / 7.5 = 14.4, rounded to 14 risers at 7.71 inches each. With 13 treads (one fewer than risers), the tread depth is 120 / 13 = 9.23 inches. The pitch is arctan(7.71 / 9.23) = about 40 degrees — a touch steep but legal. Widen the run and the pitch drops; shorten it and the stair gets steeper.
Once total rise and run are fixed, the rest of the stair follows: the step count comes from rise and target riser, the stringer length is the diagonal of the rise-run triangle, and the pitch is the per-step slope. Getting rise and run right at the start is what makes the downstream numbers fall into place; getting them wrong forces awkward compromises everywhere else.
The run you have is often dictated by the room, not by choice — a stair can only extend as far as the floor allows before it hits a wall or doorway. When the available run is shorter than the comfortable ideal, your options are a steeper straight stair, a turn (L- or U-shaped) to fold the run back on itself, or winder treads to turn within the run. Knowing your fixed total rise and your maximum available run first tells you which of those you actually need.
Total run is not just "tread depth times steps" — it is tread depth times the number of treads, which is one fewer than the riser count, because the top step lands on the floor above.
For a 14-riser stair with 10-inch treads, the run is 10 × 13 = 130 inches, not 140. Forgetting the "minus one" is the most common reason a planned stair runs longer than the space allows.
The single most important field measurement is total rise — the finished floor-to-floor height. Measure vertically from the finished surface of the lower floor to the finished surface of the upper floor, including any flooring that will be added on either level, since carpet or tile changes the number. Floors are rarely dead level, so measure at two or three points and use the largest figure; sizing to the largest rise keeps every finished riser at or under your target rather than over it.
The Blondel Comfort Rule for Stairs · How Many Steps for a Given Ceiling Height? · Laying Out a Stair Stringer