How do you develop an eye for knowing how to approach math problems or knowing what branch of maths is needed to solve something?

I'm working through coursework (pic related), and I don't know where to begin with this question or what kind of math I need to use so I can solve it. After some looking, I think I need to use polar co-ordinates to solve it but it's not something I think I would've figured out by myself - I look at problems quite literally, and if I can't immediately spot what I need to do (no matter how complex), then I'm kind of stumped.

For that question, my original idea is this: calculate the circumference (which is 6.28R) and calculate the area of the sector emerging from the centre with lines AC and CB (with angle C being 4B calculated using circle theorems), then add that value onto one hemisphere of the circle (goat side). As for the other half, I'd calculate the area of the chord AB by subtracting the area of the triangle ABT from the sector previously calculated, and add it onto the total area which should account for the curved region. However, this doesn't work because the lines AC and AB don't perfectly fall on the centrepoint C; it's slightly offset. I get the feeling I'm supposed to somehow utilise the information of the lines AT and TB being 1.5R long, but I can't think of anything. Where do I go from here? How do I develop an eye for math problems and approaching them?
I'm a first year engineering undergrad (inb4 engineers are retards), so far I've been taught polar coordinates, differential and integral calculus.

TL;DR - How do I solve math problems that aren't obvious?