The proportions of the vesica are exact and easy to check, which is worth doing, because the figure attracts a good deal of loose numerology.
Let the two circles have radius r, with centres A and B a distance r apart. The lens is symmetric about the line AB and about the perpendicular bisector of AB. Its width, measured along AB, spans a distance of r, since each circle reaches exactly r beyond the other's centre. Its height is the distance between the two crossing points. Each crossing point lies at distance r from both A and B, so it forms an equilateral triangle with them, and its perpendicular distance from AB is that triangle's altitude, r times sqrt 3 divided by 2. There are two such points, one on each side, so the full height is r times sqrt 3.
The ratio of height to width is therefore sqrt 3 to 1, roughly 1.732. That is the number, and it is not an approximation.
Two consequences follow directly. First, the crossing points and the two centres form a pair of equilateral triangles set back to back, so the vesica hands you the 60 degree angle for nothing, and six repetitions of that angle about a circle give the regular hexagon. Second, sqrt 3 is the same ratio that governs the equilateral triangle's own geometry, which is why medieval setting-out schemes working ad triangulum, by the equilateral triangle, keep producing it. The dispute recorded at Milan Cathedral in the 1390s over whether to raise the section ad quadratum or ad triangulum is the best-documented instance of masons arguing about exactly this choice.
Claims that the vesica contains the golden ratio, or encodes some particular significant number, are generally reached by measuring something other than the lens. What it demonstrably contains is sqrt 3, the 60 degree angle, the perpendicular bisector and the equilateral triangle.