My favourite trick, when I was helping students at the Physics Homework Centre at Montana State University, was to eyeball a question for a moment and say "... which is, what, 53.13 degrees or so..." without batting an eyelid. The poor students! There they were trying to figure out which way up to hold the calculator, and the guy who was meant to be helping them was rattling off the numbers to four sig fig.
It's a bad habit, and I don't expect to grow out of it soon.
The reason I could do such a feat was largely due to the professors playing into my hands. Pretty much every triangle they ever used was some multiple of the world-famous 3-4-5 triangle - the one that's always used to introduce you to Pythagoras because 32 + 42 = 52.
Once I'd spotted this common thread - and despite it being in EVERY HOMEWORK, none of the students ever did - it was a matter of moments to check that the angles were 53.13º (the bigger one) and 36.87º (the smaller one), and rattle them off the moment they showed up.
Of course, the true mathematical ninja doesn't take degrees for an answer! He would use the trick to convert such silly numbers into radians: 53.13º divided by 4 is a smidge less than 13.3, multiplied by 7 is 93.1 - so 0.93 radians; 36.87º divided by 4 is slightly more than 9.2, multiplied by 7 is 64.4, so 0.65 radians is a good estimate.