Polygons: Interior & Exterior Angles

At a glance 10 formulas

1. 1Sum of interior angles$\text{interior sum} = (n - 2) \times 180^\circ$$n \text{ sides} \Rightarrow n - 2 \text{ triangles from a corner}$
2. 2Exterior angles$\text{exterior sum} = 360^\circ,\ \text{any polygon}$$\text{interior} + \text{exterior} = 180^\circ \text{ at each corner}$
3. 3Regular polygon, n sides$\text{exterior} = \dfrac{360^\circ}{n}$$\text{interior} = 180^\circ - \dfrac{360^\circ}{n},\ n = \dfrac{360^\circ}{\text{exterior}}$
4. 4Diagonals of an n-gon$\text{diagonals} = \dfrac{n(n - 3)}{2}$$n - 3 \text{ diagonals from each corner}$
5. 5Isosceles corner triangleIn a regular polygon, two neighbouring sides and the diagonal joining their ends make an isosceles triangle; each base angle is half the exterior angle.
6. 6Regular hexagon, side aSix equilateral triangles of side $a$ meet at the centre: circumradius $a$, longest diagonal $2a$, short diagonal and gap between opposite sides both $\sqrt{3}a$.
7. 7Polygon in a circle$\text{area} = \tfrac{n}{2} R^2 \sin \dfrac{360^\circ}{n}$$R = \text{circumradius},\ n = \text{number of sides}$
8. 8Octagon from a square$s = x(1 + \sqrt{2})$$s = \text{square side},\ x = \text{octagon side}$
9. 9Whole-number anglesEach angle of a regular $n$-gon is a whole number of degrees exactly when $n$ divides 360 (with $n \ge 3$).
10. 10Angles in an APWith $n$ sides, smallest angle $a$ and step $d$, solve $\tfrac{n}{2}(2a + (n - 1)d) = (n - 2)180$, then reject any $n$ whose largest angle reaches $180^\circ$.