2D-3D

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

Spherical Triangles on Very Big Spheres

Stage: 5 Challenge Level:

Use the fact that for small values of x $$\cos(x) \approx 1 - \frac{x^2}{2} + \frac{x^4}{24}$$

to find an approximation to the identity which holds for large values of R.