by

Jim Wilson

Two larger circles with equal radii pass through each
other's centers. A smaller circle can be created inside the overlapping region so
that it is tangent to the other circles. (Tangent means that the circles touch
each other but do not cross over each other, nor do they leave any gaps.)
Compare the area and circumference of the smaller circle to the area and
circumference of the larger circle.

First,
let’s construct a GSP sketch for the problem. Take a segment AB as the radius of one of the large
circles and construct the two large circles.

Construct
the internal circle.
We know that if the circle inside is tangent at the center points, then
the radius of the small circle is one half of the radius of the large
circle.

So construct the segment connecting the two centers and take its midpoint. That is the center of the small
circle.

Click here to open a GSP file of the discussion.

Take
measures of the two circumferences and the two areas; compute their ratios.

Let r be the radius of the small circle and R the radius of the large circle. Then R = 2r.

The
ratio of the areas is

The
Ratio of the circumferences is