Internet Math Challenge
Deadline for solutions: Friday, 16 January 2004
This week's puzzle isn't too hard, but it will introduce you to a cool
geometrical object -- the Reuleaux triangle (pronounced "Roo-low").
This "triangle" is pictured in the diagram at right (the yellow shape). As
you can see, it really isn't a triangle at all because its sides aren't
straight. To build a Reuleaux triangle, start with the three vertices of
an equilateral triangle (with all three sides of length 1).
Now draw three circles. Each circle is centered at
one of the three vertices, and passes through the other two vertices. The
area of overlap between these circular disks is the Reuleaux triangle.
Have you ever wondered why manhole covers are round instead of square? The
answer is found in this very property: a square manhole cover could be picked
up, rotated, and dropped through the hole it covers. However, a circular manhole
cover won't fit through that hole no matter how it is rotated. Maybe sometime
someone will manufacture manhole covers shaped like the Reuleaux triangle!
Now for this week's puzzle: find the exact area of the Reuleaux triangle!
Be sure to simplify your answer as much as possible.
Good luck!
This shape has several interesting properties. One of them is that it is
the same width no matter which way it is rotated. (That is, it will always
just barely fit through an opening 1 unit wide). The circle, of course, has
this same property, but not many non-circular shapes do. A square (with
side-length 1), for instance, will fit through an opening of width 1 if
taken through with sides parallel to the opening. But otherwise, it won't
fit!
