Tuesday, December 16, 2008

Archimedes' Book of Lemmas, Proposition #7

Problem 646: Square, inscribed and circumscribed circles
Exercise your brain. Archimedes wrote the "Book of Lemmas" more than 2200 years ago. Solve the proposition #7 (high school level) and lift up your geometry skills.

Archimedes' Book of Lemmas #7.
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  1. Let Circle Ci's radius be r, then Circle Cc's radius is (sqrt2)r.
    Area Circle Cc=pi*2r^2, Area Circle Ci=pi*r^2, therefore Area Circle Cc=2 Area Circle Ci.

  2. if one side of the square is S1 then two half circles = S2 because if you put them together it looks like it. Also, the radius of S2 is like half of one side of a square