91Ó°ÊÓ

Verify Euler's formula for a 'pyramid' that has an \(n\)-gon as a base.

Assume that the Earth is a sphere of radius 4000 miles. Show that the spherical distance between London (latitude \(51^{\circ}\) north, longitude \(0^{\circ}\) ) and Sydney (latitude \(34^{\circ}\) south, longitude \(151^{\circ}\) east) is approximately 10500 miles.

Show that if an equilateral spherical triangle has sides of length \(a\) and interior angles \(\alpha\), then \(\cos (a / 2) \sin (\alpha / 2)=1 / 2\). Deduce that \(\alpha>\pi / 3\) (so that the angle sum of the triangle exceeds \(\pi\) ).

Let \(A, B, C\) and \(D\) be the vertices of a regular tetrahedron. Show that the midpoints of the sides \(A B, B C, C D\) and \(D A\) are coplanar, and form the vertices of a square.