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 x-axis plus the area between S and y-axis is constant!
Moreover, that constant is equal to the length of S:
A + B = s2 - s1.
In Figure 1, note that regions A and B overlap; in that
portion the area is counted twice. The quantity
(s2 - s1)
represents the length of S along the arc from
s2 to s1.
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 sector-triangle formulas from
geometry, to compute the corresponding areas.
How to Cite this Page:
Su, Francis E., et al. "Arclength Surprise."
Math Fun Facts.