I'm looking for an online tool that will let me draw lines and circles etc on a plane and then tell me the positions of any intersections, the size of any angles formed and potentially lengths of formed arcs.
Does anyone know of one?
You might be interested in GeoGebra. Is free, easy-to-use and has all features you are looking for.
This is the best free software which is very close to your request: http://car.rene-grothmann.de/doc_en/download.html