Friday, March 18, 2011

Some Pi to Celebrate St. Patrick's Day!

This four-leaf clover consists of four coplanar circles. Each circle is externally tangent to two others, as shown. The two smaller circles are congruent, and the two larger circles are congruent. A square is constructed such that each of its four vertices is also the center of one of the four circles. Suppose the area of the square is 6 1/4 in2, and the combined area of the four circular leaves of the clover is 6 1/2π in2. What is the positive difference between the radius of a large circular leaf and the radius of a small circular leaf? Express your answer as a common fraction.

Another four-leaf clover also consists of 4 coplanar circles. The large circular leaves are externally tangent to each other, as well as to each of the smaller circular leaves, which are also congruent to one another. The radius of the large and small circular leaves is 1 1/2 inches and 1 inch, respectively. What is the area of a rhombus formed such that each of its vertices is also the center of one of the four circular leaves?

Three-leaved clover

The quadrifolium (also known as four-leaved clover)

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