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Let O be the origin of the Spiral of Theodorus and P_1, P_2, P_3, ..., P_17 its vertices, where the k-th triangle has vertices O, P_k, and P_(k+1) for 1 <= k <= 16. The diameter of the spiral is the greatest distance between the vertices, corresponding to the segment joining vertices P_7 and P_17. That is, Sup{dist(P_i, P_j) | 1 <= i,j <= 17} = P_7P_17.

For simplicity, let us rename A = P_7 and B = P_17, so the diameter is equal to the longest side of triangle AOB, where OA = sqrt(7), OB = sqrt(17), and the angle AOB measures w = Sum_{k=7..16} arctan(1/sqrt(k)), since the angle at vertex O of the k-th triangle is arctan(1/sqrt(k)).

Then, by the law of cosines, it follows that the diameter d of the Theodorus spiral is equal to d = sqrt(24 - 2*sqrt(119)*cos(w)) = 6.7336560131425466317...

On the other hand, it is known that the perimeter p of the spiral (up to the 16th triangle) is p = 17 + sqrt(17) (see A373785). Thus, it is observed that p/d = 3.1369445639..., so |Pi - p/d| < 0.005.