The two solids in a poured stair
A cast-in-place stair is a stepped mass sitting on a sloped slab. The stepped mass is the part you see: a staircase-shaped prism whose cross-section is the familiar zig-zag in elevation. The sloped slab underneath - the throat - is the structural element. It is what spans from the bottom bearing to the top landing and carries everything above it, and on a self-supporting flight it is a large share of the concrete.
Estimating the steps and forgetting the throat is the standard error, and it is a large one. On the default four-step stoop, the stepped mass is 21.39 ft³ and the throat adds another 8.69 ft³ - the throat is 8.69 ÷ 30.08 = 29 percent of the pour. On a long flight the throat share falls, because the stepped mass grows with the square of the step count while the throat grows only linearly with slope length.
Not every stair has a throat. Steps cast directly on compacted fill or on an existing slab bear continuously on the ground, so there is nothing to span and the throat thickness is zero. That is the right model for a garden step or a stoop built up on hardcore. A flight rising from a path to a porch with air underneath is the other case, and there the throat is not optional - it is the beam.
Why the step count appears as n(n+1)/2
Slice the stepped mass vertically, one slice per tread. The slice under the bottom step is one riser tall and one tread deep. The next is two risers tall, the next three, and so on up to n. Each slice has area R × T multiplied by its own step number, so the total cross-sectional area is R·T·(1 + 2 + ... + n). That sum is the triangular number n(n+1)/2, which is why the volume grows roughly with the square of the step count: going from four steps to eight multiplies n(n+1)/2 from 10 to 36, so the stepped mass rises by a factor of 3.6.
Multiply that area by the width and you have the stepped volume. Everything must be in the same unit first: risers and treads are quoted in inches and widths in feet, so divide the inch dimensions by 12 before multiplying.
The throat is a slab of constant thickness lying on the slope. Its plan footprint is the total run, its rise is the total rise, and its length is the hypotenuse - L = √(rise² + run²). Volume is width × thickness × L. The thickness is measured square to the underside, which is how it is dimensioned on drawings and how you will actually set it on site with a straightedge across the sloped soffit form.
Total rise and total run come out of the same inputs and are worth checking against the site before you build a form. Total rise must land exactly on the finished landing level; if it does not, the fix is to change the riser height, not to add a lip at the top. The stair rise and run calculator handles that side of the layout.
Worked example: a four-step front stoop
Four risers at 7 in, treads at 11 in, 4 ft wide, on a 6 in throat.
- Convert. R = 7 ÷ 12 = 0.583333 ft, T = 11 ÷ 12 = 0.916667 ft, t = 6 ÷ 12 = 0.5 ft.
- Triangular number. n(n+1)/2 = 4 × 5 ÷ 2 = 10.
- Stepped area. 0.583333 × 0.916667 × 10 = 5.347222 ft².
- Stepped volume. 5.347222 × 4 = 21.388889 ft³.
- Total rise and run. 4 × 7 = 28 in = 2.333333 ft; 4 × 11 = 44 in = 3.666667 ft.
- Slope length. √(2.333333² + 3.666667²) = √(5.444444 + 13.444444) = √18.888889 = 4.346135 ft.
- Throat volume. 4 × 0.5 × 4.346135 = 8.692270 ft³.
- Total. 21.388889 + 8.692270 = 30.081159 ft³, which is 30.081159 ÷ 27 = 1.114117 yd³.
- With 10 percent waste. 1.114117 × 1.10 = 1.225529 yd³, so order 1.25 yd³. In bags: 30.081159 × 1.10 ÷ 0.60 = 55.1, so 56 bags of 80 lb mix.
Fifty-six bags is well over two tonnes of material and a long afternoon with a mixer, which is why a stoop this size is usually a short-load delivery even though it is barely over a yard.
Reading the result and setting the geometry
The volume tells you what to order; the rise and run tell you whether the stair is legal. IRC R311.7.5 caps residential risers at 7.75 in and requires treads of at least 10 in, and it also limits the variation between the largest and smallest riser in a flight to 3/8 in. That last rule matters more for a poured stair than for a wood one, because a concrete riser cannot be shimmed after the fact. Set the form once, check every riser against the same story pole, and pour.
Throat thickness is a structural decision, not an estimating one. Five to seven inches covers most residential flights of ordinary span, with reinforcement running up the slope and cross bars near the soffit, but the actual thickness and steel come from a designer once the span and load are known. A throat under about 4 in has too little section to span and too little cover to protect the steel from the weather.
Watch how the throat share moves as flights get longer. On the four-step example the throat is 8.69 of 30.08 ft³, or 29 percent. At eight steps of the same geometry the stepped mass is 77.00 ft³ and the throat 17.38 ft³, so the throat falls to 17.38 ÷ 94.39 = 18 percent. The stepped mass dominates tall flights; the throat matters most on short ones. Both are in the total either way, which is the point of calculating them separately.
Concrete for a 4 ft wide stair with a 6 in throat
| Steps | Total rise (in) | Stepped mass (ft³) | Throat (ft³) | Total (ft³) | Cubic yards |
|---|---|---|---|---|---|
| 1 | 7 | 2.139 | 2.173 | 4.312 | 0.160 |
| 2 | 14 | 6.417 | 4.346 | 10.763 | 0.399 |
| 3 | 21 | 12.833 | 6.519 | 19.352 | 0.717 |
| 4 | 28 | 21.389 | 8.692 | 30.081 | 1.114 |
| 5 | 35 | 32.083 | 10.865 | 42.949 | 1.591 |
| 6 | 42 | 44.917 | 13.039 | 57.955 | 2.146 |
| 7 | 49 | 59.889 | 15.212 | 75.101 | 2.781 |
| 8 | 56 | 77.000 | 17.385 | 94.385 | 3.496 |
Stepped mass = 0.583333 × 0.916667 × n(n+1)/2 × 4. Throat = 4 × 0.5 × n × 1.086534 ft of slope per step. Values are produced by the same expressions the calculator uses.
What goes wrong on a concrete stair pour
- Forgetting the throat. On a short flight it is close to a third of the concrete. Order without it and the last step is cast cold.
- Measuring the throat vertically instead of square to the slope. On the default geometry the slope is 32.5 degrees from horizontal, so a 6 in vertical dimension is only 6 × cos 32.5 = 5.06 in of real section - a 16 percent shortfall in structural thickness.
- Counting treads rather than risers. A flight with four risers landing on a porch shows only three full treads; this calculator wants the riser count, and the top riser's tread is the landing itself.
- Unequal risers. Concrete cannot be adjusted after the pour. Divide the measured total rise by the riser count and set every form board from one story pole.
- Under-bracing the riser boards. Wet concrete pushes hard on a 7 in board spanning 4 ft. Kickers at mid-span and a strongback on the outside face are cheap next to a blown form.
- Measuring the tread to the wrong point. If your riser boards are bevelled to form a nosing, the run that belongs in the calculator is the nosing-to-nosing dimension.
The throat carries the flight - size it with a designer
This calculator gives quantities, not capacity. A self-spanning stair is a one-way slab on a slope, and its thickness, reinforcement and bearing details depend on the span, the live load and the exposure. Reinforcement in exterior stairs also needs cover against freeze-thaw and de-icing salts. Where the flight bears fully on compacted ground for its whole length, set the throat to zero and the calculation becomes a simple stepped mass - but confirm the bearing is real before you rely on it.
Poured stairs against the alternatives
Cast-in-place is one of three ways to get a concrete stair. Precast units arrive complete and are set in an hour, which suits a tight programme and a standard rise, but they need craneage and they fix your geometry to a catalogue. Masonry steps - block or brick shells filled and capped - suit repairs and short rises where matching existing work matters more than speed. Cast-in-place wins where the geometry is bespoke, the flight is integral with a wall or a landing, or the finish must run continuously into an adjoining slab.
The quantities here feed straight into the rest of a takeoff. Reinforcement for the throat and the steps comes from the rebar weight calculator and the rebar spacing calculator. If you are batching on site rather than ordering ready-mix, the concrete mix ratio calculator converts the volume into cement, sand and stone. And if the stair lands on a slab or a footing poured at the same time, add those volumes with the concrete slab calculator before you place a single order - one delivery is nearly always cheaper than two.
Where the same level change has to be accessible as well as walkable, a ramp runs alongside the steps rather than replacing them; the wheelchair ramp slope calculator sizes that. One last practical point: pour the throat and the steps in one placement wherever you can. A construction joint between them puts a plane of weakness exactly where the shear is highest, and it shows on the finished soffit forever.