It's not surprising that rainbows have received a great deal of attention: in textbooks, in magazines, and on the web. They are, after all, beautiful, fascinating, occasionally awe-inspiring, even a little mysterious. They are an almost perfect blend of natural beauty and simple physics. Has everything that can be said about rainbows already been said? Possibly. But, pedagogically, there may be gaps. A student learning about rainbows is likely to learn that they are caused by light whose colors are dispersed by water droplets, that the dispersion occurs because the index of refraction of water depends on wavelength, that the primary rainbow results from light that is refracted into a droplet (actually, of course, many droplets) and is internally reflected once in the droplet before being refracted back out of the droplet; further, that a secondary rainbow results from a similar sequence, but with two internal reflections instead of one--and that higher-order rainbows result from more than two internal reflections. What may not be supplied in a typical presentation on rainbows are answers to the following questions (which I pose here for the primary rainbow but that could be modified for higher-order rainbows): 1. How can one use the physics of refraction to determine the angle at which the primary rainbow is seen? 2. Why is the primary rainbow apparently brighter than the sky beneath it, even though more light, after the refraction-reflection-refraction sequence, goes into the region below the rainbow than into the rainbow itself? 3. Since dispersion separates light into its component colors, why is color seen only in the rainbow, and not in the illuminated region below the rainbow? 4. About what fraction of the refracted-reflected-refracted light goes into the rainbow itself and what fraction into the region below it? In this article, I want to show how these questions can be answered using a graphical technique accessible to all students.