Comparative to most of the world, a middle class family in America is rich.
In America, they're considered middle class.
So first, some perspective. When someone has needs that are not fulfilled (food, shelter, clothing), anyone who meets needs is rich.
Okay, so that's nice. But what about US Rich?
Well, while not a perfect proxy (some rich households have no income, they are retired), household income might have some decent traction.
http://en.wikipedia.org/wiki/Household_income_in_the_United_States
I'll plunk myself on this chart. My wife and I are right at the cusp of the very last category. So our income is greater than about 98% of all households in the US.
It would be nice to adjust that for cost of living differences. 240K in Washington DC is not 240K in Atlanta or 240K in Omaha.
You certainly would not see my wife and I considered rich by strangers who might see us passing. We drive a Rav4. It's scratched. It's a few years old. Obviously used. Or we're on scooters or bicycles. Not. Rich. We live in 1/2 of a duplex that we own (well we own more of it than the bank). We shop at Costco. There's plenty of folks with enough to "live richly" but who do not (most people who "live" a rich lifestyle are not rich, at least, not for long.
So really, think about what rich really means.
I would say "rich" would be some amount of income and assets such that one can live where one wants to live, do the things one wants to do, work how one wants to work. That's different for everyone.