Figure 1

Take a circle C, and a chord in the circle.
Now slide the chord around the circle. As you do this,
the midpoint of the curve will trace out
a smaller concentric circle. Call the area between
the two circles A(C).
Now suppose you do the same thing with a larger
circle C' but with the same length chord?
Will A(C') be larger or smaller than A(C)?
Surprise: they will actually be the same area!
In otherwords A(C) does not depend on what circle C
you start with, only the length of the chord!
Presentation Suggestions:
From this fact, ask students if they can see quickly
what the fixed area must be! [Hint: start with a circle
whose diameter is the length of the chord.]
The Math Behind the Fact:
In fact, an even more amazing fact is true: take
any convex shape C and place a chord of fixed
length in it. Now slide as you slide the chord around
C, the midpoint traces out another figure D. The area
between C and D does not depend on what shape you
started with!
How to Cite this Page:
Su, Francis E., et al. "Sliding Chords."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
