three circles, each with a radius of 6 inches, are externally tangent to each other. What is the area if the region which is the exterior of all three circles but which is bounded by the three circles?
We want the area of the figure bounded by the red arcs:
Draw equilateral triangle ABC by connecting the centers
of the three circles:
Each of the interior angles of equilateral triangle ABC
is 60° or radians. So in particular angle DAF is
60° or radians.
Sector ADF has area given by the formula
where is in radians
The other two sectors CEF and BDE are congruent to sector ADF
and so they also have area , so the three sectors together
have a total area of or .
An equilateral triangle has area .
Each side of equilateral triangle ABC is , so its area
is given by
To find the area bounded by the red arcs, we subtract the total
area of the three sectors from the area of the
equilateral triangle ABC, and so the final answer is
Edwin