How high does my Ridge beam have to be for an 8x10 shed with a 4/12 pitch?

Instead of giving you the answer, lets figure it out.

Basically you start with a right triangle anf your trying to find the height h.
The base is 1/2 the dimension of usually the shortest side. (b = 4')
The length of the riser is 4"/12" * (4'*12); that's an anser in inches.
Convert that to feet by dividing by 12.; call that c

Then use the pythagorean theorm c^2=h^2+B^2

Solve for h.

KISS, you are assuming his roof is highest in the middle and slopes down to either side. But his shed could have a shed roof (lean-to) where there is just one straight roof. I also assume the rafters run across the 8 foot distance same as you did because this would give the strongest support for the money. If a shed roof 4'' for every 12" of run gives 32" higher on one side than the other. Make it easy on yourself just use whichever length of rafter comes the closest (a ten footer?) and let the rest on as an overhang at either end

The question was how high is the ridge beam, not how long is the rafter.
4/12 means 4" rise in 12" (1 ft) run.
Assuming a center ridge beam the run is 4'.
So 4" x 4 = height of ridge beam.

Which I said, 32" higher on one side than the other, both of us assuming he is spanning the shortest side. If he is going the other way out of sheer cussedness the dimension is different. And the actual height depends on how high the sidewall is to begin with that he is going upward from. I mean it's an 8x10 shed and he's got questions? I could knock one out in my sleep with no trouble. Just trying to save him some time here.

But in fact it could be a salt box style. I could use those clear polycarbonate corrugated panels I saw yesterday at L***es, face it for southern exposure, I could start transplants and tomatoes in there. It would be really nice, much better than spending that money for new living room carpet like my wife wants