This week's Challenge Question:
Given a circle of any given radius "R", and an adjustable compass that holds its size and not calibrated in any way, plus a straight edge also not calibrated or marked in any way, and the ruler and compass to be used separately. Show how to construct a square of lengths "L" that has an area that approximates the area of the circle of diameter "D". Your construct is to differ from that of Srinivasa Ramanujan construct. But your construct to have the same required precision as Srinivasa Ramanujan, ( L²/R²)∏ < 0.000000005.)
Thanks to jdretired for providing us with this puzzle.
And the Answer is...
Scribe circle C1 with centre O, scribe horizontal line WE and vertical line NS both passing through circle C1 centre O, scribe line VZ at 45 deg and line XY at 45 deg both passing through the centre of circle C1 at O, construct square VXYZ. Scribe lines S1,W1 and W1,N1 and N1,E1, divide O,W1,V,N1 into 4 equal parts by joining up the intersections, and include the diagonal bottom right and top left. With compass set to radius V,T scribe circle C2, extend line Z,V to intersect C2 at M.
With compass set to C1 radius, at centre S scribe an arc intersecting circle C1 at R1, scribe a line from O to R1 intersecting line V,Y at P. Scribe a line from P parallel to W,E to intersect circle C1 at Q. Scribe a line from Q to M, intersecting lines W1,N1 at A, line Y,X at B and the bottom right diagonal at C, with compass set to radius C,A at centre Z scribe circle C3, extend line V,Z to intersect circle C3 at R2. With compass set to radius O,R2 scribe an arc from R2 to intersect an extension of E,W at F. and an extend of line N,S at G. The distance from F to G equals L being the length of a side of a square that would approximate the area of the circle C1.
Maths:
In the maths example D = 10, therefore L using ∏ = 8.862269255.
Calculate length Y,Z. = √ ((10²)/2) = 7.071067812
Calculate length H,S2 = ((Y,Z)/8)x5) = 4.419417382
Calculate height S2,O = ((Y,Z)/2) x tan 30 = 2.041241452
Calculate S2,Q = √(R² (S2,O)²) = 4.564354646
Length H,Q = H,S2 + S2,Q = 8.983772028.
Height H,M = H,S2 + S2,O = 6.460658835.
Find angle Z,M,Q = atan(H,Q/H,M)  45 = 9.278287358 deg
Length A,B = (R/2)/cosM = 2.533141376
Note angle M = 9.278287358 deg is also a constant for this construct.
Find length L = √((D+AB)²)/2) = 8.862269257
Therefore the length L has an error of 0.000000002
Accuracy ratio (L²/R² = 3.141592655)  (∏ = 3.141592653) = 0.000000002
Therefore ratio error to pi = 0.000000002

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