A stair is a division problem

A stair looks like carpentry and behaves like arithmetic. You are handed one number you cannot change — the floor-to-floor height — and your whole job is to divide it into equal pieces that a human leg accepts.

Risers are a whole number. Always.

You can't cut 15.43 risers. So the division runs backwards from how it looks:

1. Pick a target riser — 7 inches is the comfortable default.
2. Divide the total rise by it and round to a whole number. That's your riser count.
3. Divide the total rise by that count. Now you have the real riser height, and every step is identical.

A 9 ft floor-to-floor is 108 inches. 108 ÷ 7 = 15.43, round to 15 risers, and 108 ÷ 15 = 7.2 inches per riser. Not 7. The target only chose the count; the height falls out of the count.

That last step is the whole safety story. Uneven risers are the classic stair fall — the leg calibrates on the first step and trusts the rest. Which is why you never "make up the difference" on the top or bottom step, and why the number that matters is the one after the rounding.

Treads are always one fewer

Count the surfaces you step on and you'll come up one short of the risers, every time: the top riser lands you on the floor above, and that floor is not a tread.

$\text{treads} = \text{risers} - 1$

So 15 risers give 14 treads. At a 10½ inch tread the stair eats 14 × 10.5 = 147 inches — over 12 feet of floor. That's the number that decides whether the stair fits the house, and it's the one people discover last.

The stringer is then just the hypotenuse of the whole flight: √(108² + 147²) = 182.4 inches on the slope, at 36.3° — a normal, comfortable pitch.

Two hard limits and one soft one

Residential code (IRC R311.7.5) draws two lines you don't cross:

• Riser: 7¾ in maximum. Our 7.2 passes with room.
• Tread: 10 in minimum. Our 10½ passes.

Then there's the rule no inspector enforces and every good carpenter obeys — the Blondel comfort rule, three centuries old:

$24\text{ in} \le 2R + T \le 25\text{ in}$

Ours: 2(7.2) + 10.5 = 24.9 in. Right at the top of the range.

Watch what happens if you try to save floor space with 14 risers instead of 15: the riser becomes 108 ÷ 14 = 7.71 in — still legal, just under the 7¾ cap — but 2R + T = 25.9. Legal and uncomfortable. The code sets the floor for safety; comfort is a separate, higher bar that you choose to meet.

Run it

Open the stair calculator with this flight loaded:

1. Confirm 15 risers at 7.2 in, 14 treads, and a green comfort check.
2. Drop the tread to 9 in. The tread check fails — and notice the stair got shorter, which is exactly the trade the code refuses.
3. Raise the total rise to 120 in and watch the count step to 17 risers. The riser height drops back into range on its own; the arithmetic protects you if you let it.