And 100% (1 in 1) is thus a
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
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) =
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:
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.
Does gradient matter. Yes, for one it effects your braking distance.