Figure out uncommon angles is not the easiest thing to do unless you use tools and formulas. Figuring out the angles to cut the mitres on a square is quite simple. We known that the corners on a square are 90 deg. Taking the 90 deg and dividing by 2 (45 deg.) will give you the angle to cut the mitres to form a perfect 90 deg. corner.
But when it comes to figuring the angles to cut the mitres for an octagon (8 sides), a hexagon (6 sided) or a pentagon (5 sided), it becomes more involved. I will be demonstrating how this can easily be mastered.
Figure out uncommon angles by using the circle at 360 deg. becomes quite simple. These odd shapes are nothing more than adding angles to a circle.
Figure out uncommon angles, the octagon having 8 sides tells me that we must have 8 angles in that circle and the circle has 360 deg. total.
By taking the 360 deg. and dividing it by 8 will give you the deg. of each angle of the octagon which would be 45 deg. Now to get the deg. of your mitres for an octagon, you would divide the 45 deg. corners by 2 giving you 22 1/2 deg. mitres which will form a perfect octagon.
Figure out uncommon angles, the hexagon is done in the same manner as the octagon. Take the 360 deg. circle and divide it by 6 giving you 60 deg. angles in the hexagon. to figure out the mitres, divide the 60 deg. by 2 making the mitres 30 deg. to form the perfect hexagon.
This shape is also done in the same manner as the previous shapes. Take the 360 deg. and divide it by 5 to give you a 72 deg. angles in the pentagon. By dividing this by 2 will give you 36 deg. mitres to form a perfect pentagon.
There are other ways of figuring out angles such as with a protractor or a digital protractor. These methods are all very effective and I recommend using what works for you.
Always take your time for accuracy.