How The Surveyor's Wheel Works
The surveyor's wheel is marked in fractional increments of revolution from a reference position and its current position can be represented as of a revolution from this reference, where a and b are integers. In the figure on the right, the blue line is the reference starting point. As the wheel turned during measurement, it is shown the wheel sweeps out an angle of radians or turns. In this situation, the fraction, would be the relevant ratio. The usefulness of this ratio becomes clear after further consideration of the equation for the arc length of a circle.
This equation is
,
where is the arc length, is the angle, in radians, of the circle swept through and is the radius of the circle. Now, substitute into the arc length equation the conversion from radians to revolutions to obtain the form,
.
The equation for the circumference of a circle, can clearly be seen and simplifying gives,
.
Thus showing that the base unit of measurement of the surveyor's wheel is determined only by the circumference of the wheel attached.
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