The Area of Perfect Star Shape

The Area of the perfect star shape is,

  A =\frac{1}{2} nbD \>\>\>\>(1)
Where n is a number of vertices, D is a diagonal from original o to vertex c and b is the base of triangles.

Here the perfect start shapes mean that the diagonals in all triangles to original are equal.


The area is,

A_1 =\frac{1}{2} {bh} +\frac{1}{2} {bh’}

A_1=\frac{1}{2} b(h+h’)

Where, h+h’=D, Thus,

\begin{equation} \label{eq:mak}
A_1 = \frac{1}{2} bD \>\>\>\>(2)

Then the Area of Star shape is a total of all triangles which is,

A=\frac{1}{2} {nbD}

The area for four vertices is,

\begin{equation} \label{eq:mako}
A=\frac{1}{2} (4 bD)={2bD} \>\>\>\>(3)

The area for five vertices is,

\begin{equation} \label{eq:makoo}
A=\frac{5}{2} {bD} \>\>\>\>(4)

Figure 1: Star shape with four vertices.