The Step Formula: Why 2h + r = 63 cm Matters
The step formula, known in German as the Schrittmassregel, is the foundation of comfortable staircase design. This deceptively simple equation has guided architects and builders for hundreds of years, and it remains the primary criterion for evaluating whether a staircase will feel right underfoot.
The Origin of the 63 cm Value
The number 63 cm corresponds to the average stride length of an adult walking on level ground at a natural pace. This was first systematically measured and applied to staircase design by the French architect Francois Blondel in 1675. Blondel observed that when people climb stairs, they shorten their horizontal stride and redirect energy into vertical movement. He determined that each centimeter of rise effectively consumes two centimeters of horizontal stride.
This means that if a step has a rise of 18 cm, the climber expends the equivalent of 36 cm of horizontal stride just going up, leaving only 27 cm for the horizontal run. The total energy expenditure remains equivalent to a 63 cm stride: 2 x 18 + 27 = 63 cm.
Modern ergonomic research has confirmed Blondel's observation. Studies show that staircases designed to the step formula require the least metabolic energy to climb and descend, and users report the highest subjective comfort ratings.
How to Apply the Formula
To use the step formula, you need to know your floor-to-floor height and choose either a rise or run value. The formula then determines the other.
Starting from Rise Height
If you know the rise (h), the run is: r = 63 - 2h. For example, with a rise of 17 cm: r = 63 - 34 = 29 cm. With a rise of 19 cm: r = 63 - 38 = 25 cm. With a rise of 16 cm: r = 63 - 32 = 31 cm.
Starting from Run Depth
If your available space dictates the run (r), the rise is: h = (63 - r) / 2. For example, with a run of 28 cm: h = (63 - 28) / 2 = 17.5 cm. With a run of 25 cm: h = (63 - 25) / 2 = 19 cm.
The Acceptable Range: 59 to 65 cm
While 63 cm is the ideal target, a tolerance band of 59 to 65 cm is generally accepted. Within this band, the staircase will still feel natural to most users. However, the edges of the range have distinct characteristics.
At the lower end (59 to 61 cm), steps feel slightly cramped. This often occurs with steep stairs that have a high rise and short run. The user must take smaller, more cautious steps, which can feel tiring on longer flights.
At the upper end (63 to 65 cm), steps feel slightly stretched. This typically occurs with shallow stairs that have a low rise and deep run. While comfortable going up, these stairs can feel awkward when descending because the user must reach forward for each step.
The sweet spot of 61 to 63 cm provides the most natural walking rhythm for the widest range of users.
What Happens When You Deviate from the Formula
Too Steep (Result Below 59 cm)
When the sum 2h + r falls below 59 cm, the stairs are too steep. Each step requires disproportionate effort, and the short run provides insufficient footing. Users tend to place only the ball of the foot on each tread, increasing the risk of slipping. Descending becomes particularly hazardous because the foot may overshoot the tread edge. Staircases in this range include ladders and ship's companions, which are acceptable for occasional use but not for primary residential access.
Too Shallow (Result Above 65 cm)
When the sum exceeds 65 cm, the stairs become ramp-like. The long, shallow treads disrupt the natural walking rhythm. Going up, the user must lean forward and take unnaturally long steps. Going down, the pace feels sluggish. Staircases with values above 65 cm are sometimes seen in public buildings designed for very slow traffic (such as hospital corridors), but they waste space and feel uncomfortable for normal use.
The Companion Rules
The step formula does not work alone. Two additional rules verify safety and comfort from different angles.
Safety Rule: h + r = 46 cm (Range: 44 to 48 cm)
This rule ensures that the combined rise and run provide a safe stepping geometry. It catches edge cases where the step formula might be satisfied but the individual values are extreme (for example, a very low rise combined with a very long run).
Comfort Rule: r - h = 12 cm (Range: 10 to 14 cm)
This rule checks that the run is sufficiently deeper than the rise. If the difference is too small, the staircase feels steep even if the step formula is met. If it is too large, the staircase feels like walking up a shallow ramp.
A staircase that satisfies all three rules simultaneously will be safe, comfortable, and energy-efficient. Our staircase calculator checks all three rules automatically and displays the results with clear pass or fail indicators.
Historical Context and International Variations
While the 63 cm value is standard in Germany and much of Central Europe, other countries use slightly different targets. In the United States, the International Residential Code specifies a maximum rise of 7.75 inches (19.7 cm) and a minimum run of 10 inches (25.4 cm), which gives a step formula result of approximately 65 cm. In the United Kingdom, Building Regulations Part K uses a slightly different approach but arrives at similar proportions.
The underlying principle is universal: human biomechanics dictate a consistent relationship between vertical and horizontal movement on stairs. The specific numbers may vary by a few centimeters depending on average population height and cultural preferences, but the fundamental 2:1 ratio between rise and run energy expenditure holds true everywhere.