Pentagram

January 26, 2021 Shreyas Comments 0 Comment

Fun Facts about Pentagram

Do you know that sum of all angles of a pentagram whose all five sides are extended to meet at five points is $180^{\circ}$ ?

For example, sum of all angles $x_1$ , $x_2$ , $x_3$ , $x_4$ and $x_5$ in below figure of a pentagram is equal to $180^{\circ}$ .

Proof:

It is easy to prove that:

(1) $\begin{equation*} a + b = 180 + x_1 \end{equation*}$

(2) $\begin{equation*} b + c = 180 + x_2 \end{equation*}$

(3) $\begin{equation*} c + d = 180 + x_3 \end{equation*}$

(4) $\begin{equation*} d + e = 180 + x_4 \end{equation*}$

(5) $\begin{equation*} e + a = 180 + x_5 \end{equation*}$

Let’s say sum of all angles from $x_1$ to $x_5$ is S. Adding all five equations (1) … (5) above, we get:

(6) $\begin{equation*} 2(a + b + c + d + e) = 900 + S \end{equation*}$

We already know that sum of all interior angles of a pentagon is $540^0$ . So above equation (6) becomes:

$\begin{equation*} 2 * 540 = 900 + S, therefore S = 180 \end{equation*}$

Generalization:

We can generalize this to any star polygon with number of vertices n greater than 4. It is:

$\begin{equation*} 360 (n - 2) = 180n + S \end{equation*}$

$\begin{equation*} S = 180 (n - 4) where n > 4 \end{equation*}$

As An Afterthought:

My previous article about Golden Ratio applies to pentagram also. For example, suppose $\overline{ab}$ , $\overline{ac}$ , $\overline{bc}$ and $\overline{ad}$ are lengths of line segments connecting points a-b, a-c, b-c and a-d respectively in Figure 2 shown below.