Circles and radii, counting the regions

Text

A visual model of multiplication: draw $c$ concentric circles and $r$ radii. The disc is split into a number of regions equal to the product $c\cdot r$ (with $2$ circles and $3$ radii you count $6$ regions). How many regions do you get with $7$ concentric circles and $9$ radii?

Solution

The $c$ concentric circles split the disc into $c$ rings (the central disc plus $c-1$ annuli). Each radius cuts every ring, and $r$ radii split it into $r$ sectors. In all $N = c\cdot r = 7\cdot 9 = \boxed{63}.$ Since $c\cdot r = r\cdot c$, drawing $9$ circles and $7$ radii would give the same number of regions: a geometric way to “see” the commutativity of multiplication.