Gradient: the basics

If the word “gradient” brings back horrible memories of Year Nine maths, you aren’t alone. The study of gradient is most students’ first introduction to the world of differential calculus, but cyclists need not worry – you don’t need a mathematics degree to know that a hill with a 10% gradient is steeper than a hill with a 5% gradient.

But just what does 10% mean when we are taking about roads? It means leg-aching, lung-busting hard work, but let’s see why.

If you remember back to the dark days of high school maths, you might recall the terms dy/dx or m or rise/run. These are all ways of expressing gradient – how steep a slope is – and the last one is particularly relevant for cyclists. If you ride for one kilometre (‘run’) and the road also climbs by one kilometre (‘rise’) then that’s a gradient of 100%.*

Image courtesy of Wikimedia Commons

That is, rise/run in this case would be 1000/1000 = 1, which is then multiplied by 100 to turn it into a percentage. The image to the left shows what such a hill would look like – impossibly steep.

So what then do we mean when we talk about the ’1 in 20′, the famous climb in the Dandenong Ranges National Park? Well, for every 20 metres ridden, the road climbs by one metre. So, if we wanted to find the gradient of the 1 in 20 we would divide the rise by the run, 1/20, to get 0.05 which, expressed as a percentage (multiplied by 100) is 5%.
What about Inverness Road then, a short but steep climb also in the Dandenongs? Well, it rises by 220 metres over its 2.60 kilometre length. Therefore, 220/2600 = 0.085, multiplied by 100 gives an average gradient of 8.5%. Ouch.

So why do we use a percentage gradient to describe how steep a road is, rather than an angle? Well, without going into the detailed physics behind it, the percentage gradient of a slope can be used to determine how much extra effort is required to haul a mass up it. For example, climbing a road of 15% gradient requires a force of 15% of the mass of the bike and rider, in addition to the amount of force required to ride on a flat road. Expressing gradients as angles doesn’t allow such correlation.
Gradients also sound far more impressive when expressed as a percentage – a hill with a gradient of 20% sounds far steeper than a hill at an angle of 11.3 degrees.

*This isn’t technically correct because when you climb a hill, you ride along the hypotenuse of the triangle. For roads that rise at an angle of less than 10 degrees it’s fine to take the ‘run’ as the hypotenuse as the difference between the two in the resulting calculation is quite small. As the angle of inclination increases, the difference between the run and hypotenuse increases and it no longer becomes an accurate estimate. For a (hypothetical) road of 100% gradient, the rise and run would be 1000m but the distance ridden, the hypotenuse, would be 1420m.

Thanks to Margaret Watroba for this clarification.