--description--

A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices.

Let $C(r)$ be the sphere with the center (0,0,0) and radius $r$.

Let $Z(r)$ be the set of points on the surface of $C(r)$ with integer coordinates.

Let $T(r)$ be the set of spherical triangles with vertices in $Z(r)$. Degenerate spherical triangles, formed by three points on the same great arc, are <u>not</u> included in $T(r)$.

Let $A(r)$ be the area of the smallest spherical triangle in $T(r)$.

For example $A(14)$ is 3.294040 rounded to six decimal places.

Find $\displaystyle \sum_{r = 1}^{50} A(r)$. Give your answer rounded to six decimal places.

--hints--

sphericalTriangles() should return 2717.751525.

assert.strictEqual(sphericalTriangles(), 2717.751525);

--seed--

--seed-contents--

function sphericalTriangles() {

  return true;
}

sphericalTriangles();

--solutions--

// solution required