Figure 1

Consider a unit circle, and any arc S
on the unit circle in the first quadrant.
No matter where S is placed, the area between S and
the xaxis plus the area between S and yaxis is constant!
Moreover, that constant is equal to the length of S:
A + B = s_{2}  s_{1}.
In Figure 1, note that regions A and B overlap; in that
portion the area is counted twice. The quantity
(s_{2}  s_{1})
represents the length of S along the arc from
s_{2} to s_{1}.
Presentation Suggestions:
Draw a couple of pictures with arcs of the same
length in different positions. Perhaps assign the
computation of the areas as a fun homework exercise.
The Math Behind the Fact:
Use calculus, or sectortriangle formulas from
geometry, to compute the corresponding areas.
How to Cite this Page:
Su, Francis E., et al. "Arclength Surprise."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
