Conic SectionsReference: Angle Measures and Areas
Angle Measures
A reference of exact values of sine, cosine, and tangent of angle measures.
| x° | x rad | sin(x) | cos(x) | tan(x) |
| 15 | $\frac{\pi}{12}$ | $\frac{\sqrt{2-\sqrt{3}}}{2}$ | $\frac{\sqrt{2+\sqrt{3}}}{2}$ | $2-\sqrt{\text{3}}$ |
| 18 | $\frac{\pi}{10}$ | $\frac{\sqrt{5}-1}{4}$ | $\frac{\sqrt{10+2\sqrt{5}}}{4}$ | $\frac{\sqrt{25-10\sqrt{5}}}{5}$ |
| 30 | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| 36 | $\frac{\pi}{5}$ | $\frac{\sqrt{10-2\sqrt{5}}}{4}$ | $\frac{\sqrt{5}+1}{4}$ | $\sqrt{5-2\sqrt{5}}$ |
| 45 | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | 1 |
| 54 | $\frac{3\pi}{10}$ | $\frac{\sqrt{5}+1}{4}$ | $\frac{\sqrt{10-2\sqrt{5}}}{4}$ | $\frac{\sqrt{25+10\sqrt{5}}}{5}$ |
| 60 | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| 72 | $\frac{2\pi}{5}$ | $\frac{\sqrt{10+2\sqrt{5}}}{4}$ | $\frac{\sqrt{5}-1}{4}$ | $\sqrt{5+2\sqrt{5}}$ |
| 75 | $\frac{5\pi}{12}$ | $\frac{\sqrt{2+\sqrt{3}}}{2}$ | $\frac{\sqrt{2-\sqrt{3}}}{2}$ | $2+\sqrt{3}$ |
| 90 | $\frac{\pi}{2}$ | 1 | 0 | undefined |
Areas of Regular Polygons
The following table is a reference of areas of regular polygons when the length of the side is known. A regular polygon is one whose sides are equal in length and the internal angles are equal.
| Polygon | n | Area | Appx. area |
| Equilateral triangle | 3 | $\frac{\sqrt{3}}{4}{{s}^{2}}$ | 0.433×s² |
| Square | 4 | s² | - |
| Pentagon | 5 | $\frac{\sqrt{25+10\sqrt{5}}}{4}{{s}^{2}}$ | 1.72×s² |
| Hexagon | 6 | $\frac{3\sqrt{3}}{2}s^2$ | 2.598×s² |
| Octagon | 8 | $2(\sqrt{2}+1){{s}^{2}}$ | 4.828×s² |
| Decagon | 10 | $\frac{5\sqrt{2\sqrt{5}+5}}{2}{{s}^{2}}$ | 7.694×s² |
| n-sided polygon | n | $\frac{n{{s}^{2}}}{4\cdot \tan \text{(}{\pi }/{n}\;\text{)}}$ | - |
Miscellaneous
If an n-side polygon is iterated by connecting the midpoints of the sides of the polygon with length L0, then the sum of the areas of all the polygons converges and is given by ${{A}_{\text{T}}}=\frac{nL_{0}^{2}\cos ({\pi }/{n}\;)}{4{{[\sin ({\pi }/{n}\;)]}^{3}}}$, where AT is the total area of all the polygons.