Graphing a regular pentagon

Question

I just realized that I didn't know how to graph a regular pentagon with integer coordinates... What are some possible coordinates for a regular pentagon with the uppermost point at coordinate (0,0)?

Answer 1

It's not possible to have integer coordinates for the vertices of a regular pentagon. Take two adjacent vertices, $(0,0)$ and $(a,b)$. Draw a line segment from $(a,b)$ to $(a,0)$, so that you have a right triangle with legs of length $a$ and $b$ and hypotenuse that coincides with the side length of the pentagon.

Since the sum of exterior angles of any $n$-gon is $360^\circ$, you have that the angle made by the hypotenuse and the $x$ axis is $\frac{360^\circ}{5}=72^\circ$, which means the remaining angle must be $18^\circ$.

By the law of sines (and skipping some work along the way), you have $$\frac{\sin18^\circ}{a}=\frac{\sin72^\circ}{b}\implies\frac{a}{b}=\frac{\sqrt5-1}{\sqrt{2(5+\sqrt5)}}$$ But this number is not a rational number, which means $(a,b)$ cannot be comprised of integer components.