Our staircase

Our staircase has twelve steps. Each step goes up eighteen centimetres and is twenty-five centimetres deep.

Twelve steps; each rises $18$ cm and is $25$ cm deep.

Steps $n$, rise $h$, depth $d$ (cm): $n=12,h=18,d=25$.

So the staircase rises twelve times eighteen, which is two hundred and sixteen centimetres, and runs forward twelve times twenty-five, which is three hundred centimetres.

Rise: $12\times 18=216$ cm. Run: $12\times 25=300$ cm.

Totals: $nh=12\times 18=216,nd=12\times 25=300$.

The banister follows the slope. It is longer than both numbers, about three hundred and seventy centimetres, because it is the long side of a right-angled triangle. The steepness is eighteen over twenty-five, which is about thirty-six degrees.

The banister is the long side of a right-angled triangle: $\sqrt{216^2+300^2}\approx 370$ cm, longer than both. Slope: $\frac{18}{25}=0.72$, which is about $36^{\circ }$.

Banister $B$: $B=\sqrt{216^2+300^2}\approx 370$. Slope $\frac{h}{d}=0.72$, so $\theta =\arctan \frac{h}{d}\approx 36^{\circ }$.

The surprise is in the details. All twelve steps must be exactly the same height. If one is only two centimetres higher, your foot arrives too early and you trip, although you hardly notice the difference.

All steps must be equal. A step of $20\ne 18$ trips you, although the difference is only $2$ cm.

For all $i$: $h_i=18$. A step $h_k=20\ne 18$ trips.