To affect a quadrature of a plane shape is to construct a square with the same area. For this reason, the task is often referred to as squaring the shape. One of the most famous problems of antiquity is the problem of squaring the circle. According to the Greek tradition, the construction must be carried out with ruler and compass, which, by now, is known to be impossible.
Some quadratures are pretty obvious even to young children, especially those that are equidecomposable with a square. The simplest examples are served by the solution to the Twelve Matchsticks Area Puzzles, the executioner's ax below
Let R be the radius of the big circle, from which we cut two equal circular segments (S and T) with central angles 90o and a circle C which touches the big circle and the segment T.
The altitude in the isosceles ΔAOB equals R/√2. The height of either S or T is then
R - R/√2,
and that of the lenses the form together is double that:
2R - R√2.
Subtracting this from the diameter 2R of the big circle gives the diameter of the small circle: R√2. The area of the small circle then is πR²/2. The area of segments S and T is the difference of the area of the circular sector (πR²/4) and the triangle (R²/2). As the result, the area of the claws is:
πR² - 2(πR²/4 - R²/2) - πR²/2 = R².
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