A regular polygon in the plane is a closed figure (polygon) composed of equal length sides and all equal angles. The equilateral triangle is the regular polygon with fewest sides. As the number of sides increases to infinity, the regular polygon begins to resemble a circle. Using some known properties of the circle, we can find a formula for the number of degrees in the interior angles of any n-gon. Figure 1 shows how to get the central angle made by connecting each vertex to the center and how to get the equal side lengths.

Because each triangle is isosceles, the corner angles of each “triangle” are given by $\beta=\frac{180^{\circ}-\alpha}{2}=\frac{\pi-\alpha}{2}$. Since each angle of the n-gon is made from $2\cdot\beta$, $$\text{n-gon angle}=\gamma=\pi-\alpha.$$

If we let the center point be at the origin, $(0,0)$, and we compute $\alpha$ as in Figure 1, then, for the $7$-gon, each point is given by $$B=\left(\cos(\alpha),\,\sin(\alpha)\right)$$ $$C=\left(\cos(2\alpha),\,\sin(2\alpha)\right)$$ $$D=\left(\cos(3\alpha),\,\sin(3\alpha)\right)$$ $$E=\left(\cos(4\alpha),\,\sin(4\alpha)\right)$$ $$F=\left(\cos(5\alpha),\,\sin(5\alpha)\right)$$ $$G=\left(\cos(6\alpha),\,\sin(6\alpha)\right)$$ If the polygon side length is a given value, then for example, with $S=100$, we would first need to calculate the radial length, which is the isosceles leg length,$\ell$. $$\text{From the law of cosines }\ell^{2}+\ell^{2}-2\cdot\ell\cdot\ell\cos(\alpha)=S^{2}$$ $$\ell=\sqrt{\frac{1}{2}}\cdot S\cdot\sqrt{\frac{-1}{\cos(\alpha)-1}}$$ then, $A=(\ell,1)\quad B=\ell\left(\cos(\alpha),\sin(\alpha)\right)\quad C=\ell\left(\cos(2\alpha),\sin(2\alpha)\right)\quad etc$.