We want to determine which is the inner angle between two segments that connect the centre of the icosahedron to the centre of two adjacent triangles.
In order to do that, we observe the kite formed by the points :
In the yellow triangle, we observe that the segment is the line from the side of triangle to the centroid and is equal to where, for an equilateral triangle, . We thus know that:
We also observe that is the line from the centre of the icosahedron to the centroid of the triangle, which is equal to the radius of the inscribed sphere :
Now, for the triangle we apply :
and extract :
For triangle , we apply and substitute :
To find out the numeric value, we substitute and obtain :