2008-15.md

course	course_year	question_number	tags	title	year
Geometry	IB	15	IB 2008 Geometry	2.II.12G	2008

Show that the area of a spherical triangle with angles $\alpha, \beta, \gamma$ is $\alpha+\beta+\gamma-\pi$. Hence derive the formula for the area of a convex spherical $n$-gon.

Deduce Euler's formula $F-E+V=2$ for a decomposition of a sphere into $F$ convex polygons with a total of $E$ edges and $V$ vertices.

A sphere is decomposed into convex polygons, comprising $m$ quadrilaterals, $n$ pentagons and $p$ hexagons, in such a way that at each vertex precisely three edges meet. Show that there are at most 7 possibilities for the pair $(m, n)$, and that at least 3 of these do occur.