Hughes Walker Solicitors Ltd

The gradient of the slope of a road can be expressed as the angle of the incline ie a flat road is 0°, a perpendicular face being 90°, however traditionally, angles have never been used in the UK to describe the steepness of a road. Instead gradients of roads are generally expressed in one of two different ways, one being the "old style" and the other being the  "modern style":

1. The Old Style

The standard of unit of measure relates the number of inches, feet, yards or miles travelled vertically (the vertical rise) with the number travelled horizontally.

In other words if you climb 10 yards over a 30 yard horizontal distance that hill is a "1:3" or "1 in 3"- in other words a steep incline.

2. The Modern Style

Basically it is exactly the same concept except we are now measuring the vertical rise against every 100 units of measure along the horizontal axis. Put another way, if  you rise 33.33 yards in height over a horizontal distance of 100 yards, the gradient is 33.33% or in other words the same as 1 in 3.

This is what a 10% gradient looks like:

And 100% (1 in 1) is thus a 45° angle:

It follows from the above that it is quite difficult to express the gradient of sheer face in percentage terms (its an infinite %).

Amazingly, a Landrover, if properly driven can tackle such a 100% (1 in 1) gradient.

Obviously a gradient is the same whether going up or down the slope, but the downward gradient is designated by a minus sign ie -10% refers to descending down a 1 in 10 incline.

The clever bit about percentage gradients is that in physics terms it measures the additional (or reduced) percentage force required to move an object (eg a vehicle) up (or down) the slope in question compared to the force required to move it on a horizontal plain. Something cyclists take a keen interest in.

To put gradient into perspective, the most that a horse pulling a full load can manage is about 1 in 30 (3.3% gradient). One of the steepest roads in the UK is Hardknott Pass in the Lakes, which in places has a gradient of  about 30%, this is probably the most you should sensibly attempt in a normal saloon car. Chimney Bank at Rosedale Abbey, N Yorks is 1: 3. There is however a 300 yard stretch of  road near Harlech Castle which apparently boasts a 1: 2.5!

A third related way of expressing gradient is as the decimal fraction:  Gradient = Height / Distance. (In trigonometry this decimal fraction is referred to as the tangent of the angle of the slope)
.......and to convert the tangent of the angle to a percentage, simply multiply by 100.

 Example: 1 in 3 hill =  1/3 =  0.3333  = (0.3333 x 100) = 33.33%. Or take the same slope downwards,  dropping 1 foot in height for every 3 feet traversed horizontally = 1 / -3 =  -0.3333 = (-0.3333 x 100) = -33.33%. Thus any gradient expressed as "y in x" can very easily be converted to a percentage gradient. NB In this example, the value 0.3333 is the tangent of the angle.

A fourth way? Instead of dividing the height risen by the horizontal distance along the base (the tangent of the angle of the slope), sometimes the ratio is taken as the height risen by the length of the slope (height / length of the hypotenuse), in trigonometry this is referred to as the sine of the angle. For obvious reasons it is easier to work out the length of a slope (the hypotenuse) than the base horizontal distance. The values of the sine and tangent of a slope are often quite similar however to our knowledge the convention is ALWAYS to express a road gradient by reference to the tangent of the angle and never its sine!

 Can I work out the angle (q) of the slope from its percentage gradient? Answer: Yes, but to do so involves delving into some tricky trigonometry. (Don't worry too much, we will cheat and use an on-line scientific calculator). NB: this only works for gradients up to 100% (45° angle). Step 1: convert the % gradient back to the tangent (TAN) of the angle by dividing by 100, that bits easy! Step 2: Next convert  the value of TAN to the angle of the slope  using the inverse function of TAN. The inverse function of TAN is referred to as ARCTAN, or sometimes ACTAN or ATAN, and in mathematical annotation is written as: tan-1(x) where x is the value of TAN Unfortunately working out the arctangent by hand is fiendishly difficult, involves something called Gregory's formula and should only be attempted by higher mathematical types. Step 3: so cheat! Click on this scientific calculator then key in the TANgent number (eg for a 100% gradient, key in "1"), now switch on the inverse function of the calculator by clicking the "INV" button, then click the "TAN" button (which in the inverse function gives tan-1 ). Hey presto! The numerical answer given by the calculator is the ANGLE of the slope; so for example if you keyed in "1" you got back the answer q= 45°. Here-endeth the trigonometry lesson.

How is the gradient (tangent) of a slope actually measured?  One of the simplest techniques is to use a beam of say 100 inches in length, one end is rested on the road surface, the beam is kept perfectly horizontal with a spirit level, then the vertical drop at the other end of the beam to the road surface is measured, for example a 10 inch drop will represent a gradient of 10%. Of course this only gives a snap shot of the gradient at the point where the measure is taken, and does not give the overall gradient of the hill.