        .      =        ..=      =   . .=   ..+  x                x    +      :     
                                                                   +      :       + 
 +:        :x                    x+.      .      .  +            x     .   x=  x    
  x .  =          .            :  x   :      =         +    +   = .        =        
              : =     = .           ..       .    + .       x                     =x
    .     +    =  x                .                                                
            :    :         :                    :    :    +x            :        =  
x       :   +      . .                                         x       x   +        
                   :                x     =    :               .x        x      : : 
 :            .    = = .       +   =    +                     x            +   .    
        =  :          = :           = :         .                          :        
     .                            : =      =                 = +             x      
x     =            +     x      :            x                  :                 x 
                  +x      x                   x   .  :         .    .       =       
 x                   :     :               +  +                              .      
                            .:                    + x       =  :     x       x +    
.    :=     .     .  x =                :      + =             +      +=   ==  x    
           :                       :    x        .   x  +           +    x         x

Stair Formulas

The exact equations Stairs Calc uses — riser height, tread run, stringer length, stair angle and the comfort rules — each with a derivation and a worked example.

Number of risers: N = round(total rise ÷ ideal riser)
Round to the nearest whole number, then divide the rise back by N for the exact, equal riser height.
Riser height: riser = total rise ÷ N
Treads: treads = N − 1
The upper floor acts as the top step.
Total run: total run = (N − 1) × run
Stringer length: stringer = √(total rise² + total run²)
The hypotenuse of the staircase triangle (Pythagoras).
Stair angle: angle = atan(rise ÷ run)
The pitch of one step.
Minimum stairwell opening: opening = ((floor thickness + headroom) × total run) ÷ total rise
Throat: throat = board width − (rise × run) ÷ √(rise² + run²)
Keep at least 3½″ of solid wood behind the notch.
Comfort — Blondel: 2 × rise + run = 24″–25″
The classic stepping-cadence rule.
Comfort — Rise + Run: rise + run = 17″–18″
Comfort — Product: rise × run ≈ 70–75 in²

Worked example — a 9′-7″ total rise

Dividing by an ideal 7" riser gives 16 risers, so each riser is 7³⁄₁₆". With a 11" run there are 15 treads, a total run of 13′-8¾″, a stringer length of 16′-9″, and an angle of 33.2° — very comfortable and within IRC limits.

Written by the Stairs Calc editorial team. Methodology and code references: see our methodology.

Built and maintained by builders, drafters and engineers who plan stairs for a living — every code limit is transcribed from the published standard and cited to its exact section.

Last reviewed 2026-06-20 against IRC 2021/2024

Stairs Calc gives accurate geometry and checks it against published building-code limits, but results are estimates for planning. Codes are adopted and amended locally and change over time. Always confirm dimensions against your local adopted code and a licensed professional before you build.