What the Stair Calculator actually computes
You can describe an entire residential staircase with one measurement and two design choices. The measurement is the total rise — finished floor of the lower level to finished floor of the upper level. The choices are a comfortable riser height (most people pick something between 7 and 7.75 inches) and a tread depth that pairs with it.
The Stair Calculator takes that single input, picks a riser count that gets you close to your target riser height, applies the 2R + T comfort rule to set tread depth, and returns the rest of the geometry: exact riser height, tread depth, total run, and stringer length. It also flags whether your stair meets the International Residential Code limits — max riser 7¾", min tread 10" — so you know in advance whether the inspector will sign off.
The whole calculation rests on one constraint that catches first-time builders off-guard: risers come in whole numbers. You can't have 13.4 risers. You either have 13 or 14, and the choice of integer changes the exact riser height, which changes the tread depth, which changes the stringer length. The calculator handles the integer rounding so you don't end up cutting fourteen stringers for what should have been thirteen risers.
How to use the Stair Calculator
- Measure the total rise. Vertical distance from the top of the finished floor on the lower level to the top of the finished floor on the upper level. Use a long tape pulled tight. For an unbuilt second floor, add the joist depth and subfloor thickness to the wall height.
- Set a target riser height. 7.5" is the default — most people find it comfortable. 7" feels slightly easier but eats more floor space below. 7.75" is the steep edge of code-compliant.
- Read the riser count. The calculator divides total rise by your target and rounds to the nearest integer.
- Read the exact riser height. Total rise ÷ rounded riser count. All risers in a code-compliant stair must be this exact height (±1/4" tolerance between any two).
- Read the tread depth. Computed from the 2R + T rule with a target sum of 25 inches.
- Read total run and stringer length. Total run = (riser count − 1) × tread depth. Stringer length = √(rise² + run²).
- Check the code-compliance flags. The tool tells you whether the geometry passes IRC and whether it passes the 2R + T comfort range.
Worked example: a 9-foot floor-to-floor with a 7.5" target riser
This is the most common scenario in US residential construction. Total rise from finished floor to finished floor: 9 ft 0 in = 108 inches. Target riser height: 7.5".
- Riser count (raw): 108 ÷ 7.5 = 14.4 risers. Can't have 0.4 of a riser; round.
- Riser count (rounded): 14. (Could also be 15, but 14 keeps the riser height closer to the target.)
- Exact riser height: 108 ÷ 14 = 7.714" per riser. (Often written as 7-23/32".)
- Tread depth from 2R + T = 25: T = 25 − (2 × 7.714) = 25 − 15.43 = 9.57". That's below the IRC minimum of 10", so bump tread depth up to 10" and accept that the comfort sum becomes 25.43".
- Total run: (14 − 1) × 10 = 130 inches = 10 ft 10 in. That's how much horizontal floor space the stair eats.
- Stringer length: √(108² + 130²) = √(11,664 + 16,900) = √28,564 = 169" ≈ 14 ft 1 in. Order a couple of 16-foot 2×12s and you'll have enough for two stringers plus a third for the middle.
The code check passes: riser 7.714" is under the 7.75" max, tread 10" hits the 10" min, and 2R + T = 25.43" is just outside the ideal 24-25" range but inside any practical interpretation of "comfortable." This stair will pass inspection and feel right to climb.
For comparison: try 13 risers on the same 108" rise. Riser height becomes 108 ÷ 13 = 8.31" — over the IRC max. Inspector says no. You'd have to either drop the lower floor (impossible) or split the stair into two flights with a landing (extra cost, extra footprint). 14 is the right answer.
The 2R + T rule, in plain words
2 × riser height + tread depth = 24 to 25 inches.
It's an ergonomic rule of thumb. The math reflects the average adult stride on stairs — the relationship between how high your foot lifts and how far it travels horizontally per step that feels natural.
The rule has been around since the 17th century, attributed to French architect François Blondel. It's been validated repeatedly. Stairs in the 24-25 range feel right. Below 24 feel cramped — your feet land too quickly, you trip going down. Above 25 feel strung-out — you have to lengthen your stride on every step, which gets tiring on a long climb.
The IRC code does not require 2R + T compliance. You can build a stair with a 7.5" rise and a 14" tread and the inspector will sign off because both numbers meet the individual limits. But 2 × 7.5 + 14 = 29 — way over the comfort range. Walking up that stair feels like climbing a slow ramp. Every architect's checklist includes 2R + T because code-legal and comfortable are two different bars.
IRC code requirements at a glance
| Spec | IRC residential limit | Notes |
|---|---|---|
| Maximum riser height | 7¾" (7.75") | All risers within ±¼" of each other |
| Minimum tread depth | 10" | Nose-to-nose, not including overhang |
| Tread overhang (nosing) | ¾" to 1¼" | Required if tread is under 11" |
| Minimum stair width | 36" | Above the handrail; 31½" if handrail one side, 27" if both |
| Minimum headroom | 6'8" (80") | Measured vertically from any nosing |
| Handrail required | 4 or more risers | Height 34-38" above tread nose |
| Open-riser gap | Less than 4" | Can't pass a 4" sphere through (kid safety) |
| Landing required | Over 12 ft total rise | Minimum landing 36" deep |
Local codes can be stricter than IRC. Some jurisdictions require lower maximum risers (7"), wider treads (11"), or non-slip nosing. Check with your local building department before cutting stringers. Permit drawings catch most issues; in-progress inspections catch the rest.
Typical staircase geometry by floor-to-floor height
Most US residential floors land at one of three common total rises. Here's the math at each, with a 7.5" target riser:
| Floor-to-floor | Risers | Riser ht. | Tread | Total run | Stringer |
|---|---|---|---|---|---|
| 8'0" (96") | 13 | 7.385" | 10" | 10'0" | ~12'6" |
| 8'6" (102") | 14 | 7.286" | 10" | 10'10" | ~13'7" |
| 9'0" (108") | 14 | 7.714" | 10" | 10'10" | ~14'1" |
| 9'6" (114") | 15 | 7.600" | 10" | 11'8" | ~14'10" |
| 10'0" (120") | 16 | 7.500" | 10" | 12'6" | ~15'9" |
Notice two patterns. First, the riser height never lands exactly at the target — it lands at whatever integer divides the total rise cleanly. Second, total run grows faster than total rise because every extra riser adds one tread (10"), and the stair already has more horizontal feet than vertical at typical residential pitches. A 10-foot ceiling needs nearly 13 feet of horizontal floor space, which is why basement stair openings always feel larger than people expect.
The stringer: where Pythagoras shows up again
The stringer is the diagonal piece of lumber that holds the treads and risers along each side of the stair. It's typically cut from 2×12 (1½" × 11¼" actual dimension) and the cut pattern is a sawtooth — alternating notches for risers and treads down the length of the board.
Stringer length = √(total rise² + total run²)
Same Pythagorean theorem the Roofing Calculator uses for pitch, and the Pythagorean Theorem Calculator handles directly.
For our 9-foot example: √(108² + 130²) = 169". A 2×12 comes in 8, 10, 12, 14, 16, and 20 foot lengths. You need at least 14 ft (168") plus a few inches for the angled cuts at top and bottom, so order 16-footers and have lumber left over. Order at least two stringers (one each side) and typically three for stairs wider than 36" — the middle stringer keeps the treads from bouncing under load.
The cut math on the stringer itself is the same rise and run, just repeated. Use a framing square with stair gauges set at the riser height and tread depth, walk it down the board, mark each notch, cut with a circular saw and finish the inside corners with a handsaw. The first stringer is the template for the rest.
Open stairs, closed stairs, and outdoor stairs
The geometry is identical across all three; only the finish and material change.
- Closed stair — every riser face is covered by a vertical board. Standard interior construction; reduces noise transmission and hides the under-stair void.
- Open stair — no riser face, you can see through the gaps. Looks modern but the gap-width code applies (less than 4" between treads). Often paired with a thicker stair tread (1.5" or more) since the tread becomes a more visible design element.
- Outdoor / deck stair — same code, same math, different material. Pressure-treated 2×12 stringers or PVC composite. Riser heights for outdoor are often 6-7" rather than 7-7.5" because the use case includes mud, snow, and ice; shallower stairs are safer outside.
Some local jurisdictions require enclosed risers under stair landings (no triangle gap at the bottom) for fire-rating reasons. Worth checking before you frame the stair opening.
What the calculator doesn't handle
- Spiral stairs. Different code (minimum 7.5" tread at 12" from the center column, minimum 6'6" headroom, minimum 26" tread length). The 2R + T rule still applies for tuning comfort, but the calculator's straight-stair math doesn't.
- Curved stairs. Plan as a series of trapezoid treads where the outer edge is wider than the inner. The riser count and 2R + T math still help you size the geometry; the linear stringer formula doesn't apply.
- L-shaped or U-shaped stairs with landings. Calculate each flight separately. Landings count as one tread of full stair width (36"+) and have their own headroom requirements.
- Winders (pie-shaped treads that turn the stair without a landing). Allowed by IRC but with their own minimum-tread-depth rules (10" at the walk line, which is 12" out from the narrow edge). Treat as a custom calculation, not a straight-stair problem.
Related calculations
- Pythagorean Theorem Calculator — direct access to the math behind stringer length.
- Roofing Calculator — another rise/run problem with the same triangle underneath.
- Square Footage Calculator — for sizing the stair opening in the upper floor.
- Concrete Calculator — for outdoor concrete steps or a basement landing.
Frequently asked questions
What's the 2R + T rule and where does it come from?
2 × riser height plus tread depth should equal 24 to 25 inches. It captures the relationship between how high your foot lifts and how far it travels horizontally per step on a stair — values in the 24-25 range feel natural to climb. The rule is attributed to François Blondel in 1675 and has been validated by every ergonomic study since.
What are the IRC limits for residential stairs?
Maximum riser 7¾", minimum tread 10" (nose-to-nose, not counting overhang), all risers within ¼" of each other, minimum width 36", minimum headroom 6'8", handrail required for 4 or more risers. A code-compliant stair passes inspection; a non-compliant one can't be permitted.
Why does the calculator round the riser count?
Because risers come in whole steps. You can't physically build 13.4 risers — you build 13 or 14, and the choice changes the exact riser height (total rise ÷ riser count). All risers must be the same height in a code-compliant stair, so picking the integer happens before any cutting starts.
What if my floor-to-floor doesn't divide cleanly by 7.5"?
It almost never does. The calculator picks the integer riser count that gets the exact riser height closest to your target while still meeting IRC. For a 108" rise, 14 risers gives 7.714"; 15 risers gives 7.2". Either passes code; 14 sits closer to the typical comfort zone.
How is stringer length computed?
Pythagorean theorem: √(total rise² + total run²). The stringer is the diagonal lumber that holds the treads, and the diagonal of a right triangle is the hypotenuse. A 108" rise with a 130" run gives a stringer length of √(11,664 + 16,900) = 169" = about 14 ft 1 in.
Can I use this for an outdoor deck stair?
Yes. The geometry and code are identical for indoor and outdoor stairs. The differences are material (pressure-treated lumber or composite for outdoor) and a slight preference for shallower risers (6-7") on exterior stairs because of mud, snow, and ice. The math doesn't change.
What's the difference between rise and riser?
Total rise is the full floor-to-floor height — typically 96-120" in residential. Riser is the vertical face of one step. For an N-riser stair, total rise = N × riser height. Run and tread follow the same naming pattern: total run is the full horizontal length, tread is the depth of one step.
Why is there one fewer tread than risers?
Because the upper floor IS the last tread. You walk up the last riser and step onto the floor above, not onto another tread. Total run is therefore (riser count − 1) × tread depth, not riser count × tread depth. A 14-riser stair has 13 treads between floors.