I have a leak from the hot water pipes under the floor.

[Madrab]

Do they need to be cut down lengthwise to go around the pipe until it hits the joist?

The best way is to feed them onto the length of pipe before laying, that way you don't need to cut them. If the pipe is in and live and the lagging is a retrospective job then yes. They usually come with a pre marked cut line - they just need started with a stanley then run a thumb along the line and they should just split.

Then tape back up along the split with the stickiest tape you can find, personally I use gorilla tape as it seems to last for a long time, other tapes are obviously available.