Proper drain pipe size

It all depends oh how much water you need to get rid of and how many leaves dead gophers and snakes have accumulated in the pipe. For a straight run here ya go, it's a start ..

Do you live in a climate where temps go and remain below periods for longish (more than two weeks) periods? If so, you'll need a wider diameter pipe to handle the occasional ice buildup inside the drain pipe. Same for if your area gets substantially heavy rains at times. If you've got significant roof area, you'll have a lot of run off, and that has to be factored in also.

I live in a small house in Minnesota and ran two french drains from my downspouts. The longest starts in the back, runs along the side and connects to the downspouts at both corners, then runs downhill until the end peeps out of the ground - about a fifty foot run in total. I used a four-inch diameter pipe, but it couldn't handle all the runoff from both downspouts during the very heaviest rains, so I ended up running a new drain from the front downspout. I have a third french drain from the downspout on the opposite corner, again with a four-inch drainpipe, about an eighteen foot run, and it's worked out fine.