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From the Fun Fact files, here is a Fun Fact at the Medium level:

Arclength Surprise

Figure 1
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 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.

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 sector-triangle formulas from geometry, to compute the corresponding areas.

How to Cite this Page:
Su, Francis E., et al. "Arclength Surprise." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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References:
    Putnam examination, 1998.

Subjects:    geometry, calculus, analysis
Level:    Medium
Fun Fact suggested by:   Francis Su
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