To quote from the great SNL philosopher, Mango, "Can you touch a rainbow? Can you put the wind in your pocket? No! Such is Mango." I think he has it right. I don't know much about the optics of rainbows, but I'm pretty sure they move relative to the observer, so they do not have an objective diameter.
At least that's what I thought until I found this answer on the magical internet here:
"It's probably not impossible, but it is difficult. A rainbow looks circular because it's basically the circle where a cloud of rain droplets intersects with your cone of vision, like the circle on the end of an ice-cream cone. Imagine said ice-cream cone with the point in your eye (don't actually try this experiment unless you're looking for a career in piracy). Now make the cone bigger and bigger until the round end hits the cloud of raindrops that are reflecting the sun's light. The big circle on the end of that cone is where the rainbow appears to be -- as someone else pointed out, you can only see the top half of it (because the other half is below the surface of the earth). The raindrops reflect light at about a 40 degree angle, so you can calculate the diameter of the circle if you also know the height of the cone (because the height of the cone, the radius of the circle, and the 40 degree angle are all part of a right angled triangle). The challenge is knowing the height of the cone, which is how far away the cloud of raindrops is from you. If you can work that out then, yes, you can measure the diameter of the rainbow (diameter = (2*distanceToCloud) / tan 40)."
Also, if the ends of a perfect rainbow appear to touch the earth at two known landmarks (say, two buildings), perhaps the diameter is the distance between those landmarks?