Looking to replace my hydraulic disc brakes but intend to use the old discs. How do I tell the size of the discs to ensure I purchase the right brakes?
The disc that you are talking about is in fact called the rotor.
Rotor size is measured by diameter, ie. edge to edge through the centre. The size may also be etched on to the rotor which would be more accurate than measuring yourself. There are a number of common IS (International Standard) sizes of rotor from 140mm through to 203mm.
Callipers only come in one size but use various adapters/spacers to fit the rotor. Once you know what rotor you have you purchase a new brake set which may or may not include the adapter you need. Your current brake callipers would use the right size adapter for the rotor you are using, but may not be right for the brake you wish to purchase, companies have proprietary systems for attaching the calliper such as Avid's CPS which require special adapters or spacers.
Additionally there are two types of mounts for callipers post and IS. Most callipers are made to use post mounts while most frames still have IS mounts on the real dropout. An IS mount is like a fin with two holes in it while a post mount is two posts. IS adapters are used to allow the calliper to be fastened to the bike. Most recent forks are post mount.
As DWGKNZ says you could measure these. Alternatively, you could search online for your bike make/model & look at the specification. This would save you the time of taking the wheel off and measuring (and note front/back might be different sizes). This assumes that it has not been changed since you bought the bike.