Expert: Scott A Wilson - 11/6/2010

Question
What is the formula to determine distance traveled,
using the diameter of a wheel. A wheel with a 12"
diameter will travel a distance of 25.12' in 8
revolutions. But a wheel with a diameter of 24" will
travel 50.24' in 8 revolutions. Is there a formula for
multiple revolution problems, and what is the rule
with diameters greater than 12"?

Answer
The circumference c of a circle is 2*pi*r.
Since d = 2*r, it is also d*pi.
When n revolutions are travelleds), the wheel goes n*d.
The units must always be kept the same.  Note that 1' = 12".
Using pi as 3.1416 (and really it has infinite digits), I get
the diameter d = 75.3982".

In 8 revolutions, this is 603.1858".
That is 50.27'.  Note this is slightly different.  That is since I kept all of the accuracy in Excel (there was no rounding in the middle).  For 24", I get 110.5" = 8.38'.

The formula is space travelled s = 2*r*pi*t where pi is 3.14159265..., r = radius, and t = time.

If the diameter d is given, replace with 2*r since d = 2*r.
The distance is sometimes d, but here, d is used for diameter,
so think of the distance as space covered.

The formula is then s = c*t = pi*d*t = 2*pi*r*t
where s is the space covered (distance), c is the circumference,
pi is 3.14159265..., r is the radius, and t is the time.