First off, thanks for all the input! I'll have to check the links when I have more time.

Ah, right, I should have remembered getting my planetary mass with a pendulum as long as I know G and have found the planet's size.

Is there another method you could use to measure the planet's circumference? Not that the shadow isn't accurate or primitive enough, but still. I feel like you could turn the 'how far is the horizon' calculation around but it seems you'd need really precise methods (or large objects) to get good results from them.