### Joy of Problem Solving | Brilliant Math & Science Practice Problems

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We find a problem involving dividing a square into two areas and comparing the areas. A geometrical solution there is convincing (the blue and white regions have equal areas), but when I attack the same problem with a more cumbersome algebraic approach, I keep getting a different answer. Thus (see the website) :

The white region is the two semicircles minus the lens that has been counted twice. Call the side of the square s and the lens L. We have

W = 2 (½) pi (s/2)^2 - L

Since the whole square can be seen as four of those semicircles minus four of those lenses, we have

s^2 = 4 ( ½ ) pi (s/2)^2 - 4 L

Solving this last for L, we have

L = (pi/8 - 1/4) s ^2

Plugging this into the first equation, we have

W = (pi/8+ 1/4) s^2

The Blue region must be

B = S^2 - W = s^2 - (pi/8 + 1/4) s ^2 = (3/4 - pi/8)s^2

This is not equal to W. What is my error?

J