## Spherical triangles

### Problem 332

Published on Sunday, 10th April 2011, 10:00 am; Solved by 372; Difficulty rating: 50%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 centre (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 __not__ 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 A(`r`). Give your answer rounded to six decimal places.