The first problem could be solved more easily than was shown. If you remove the two circles from the rectangle then the red region below the diagonal is the same as the remaining area above the diagonal, so the red area is just half of rectangle minus two circles.
Do the Chinese learn trigonometry in sixth grade? The speaker should have mentioned that you need a calculator to solve the second problem, since that is the only way of calculating arc tan ½. Given enough time, I think I might have been able to solve the second problem. It sure did require a lot of work.
You might like videos by Arthur Benjamin. Do a google search and you will see a bunch.