Squaring

An ancient problem is the squaring of circle. That is construct (with compass and straightedge) a square with same area as a given circle.
This problem stayed unsolved for centuries until finally proved as being impossible. It is an equivallent problem to construct a segment with length π.
A number is constructible with compass and straightedge only if it is solution of some specific algebraic equation. But π is transcendant (over Q), that is is solution of no algebraic equation with coefficients in Q.
But some cranks still publish methods for squaring the circle, philosopher's stone and perpetual motion...

Although it is impossible exaclty, there are approximate constructions, based on approximate values of de π.
The most famous approximatie value is π ≈ 22/7 = 3.1428... instead of 3.141592...

A possible construction with that value might be :
EB = R/4, AF = AB, OP parallel to EF, AC = AP.
AE = R + 3R/4 = 7R/4
AC = AP = AO × AF/AE = R × 8/7
Then BC = R × (2 + 8/7) = R × 22/7.
Circle with diameter BC and perpendicular in O to BC intersect in D. BD is the side of the equivallent square : BD² = BO×BC = R² × 22/7

Relative inaccuracy is (22/7 - π)/π ≈ 0.0004 that is less than 1 in thousand (practical construction errors are even worse !).

 

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