The area of the triangle whose vertices are represented by the complex numbers 0,z,zeiα,(0<α<π)  equals:

Vertices are 0=0+i,0,z=x+iy and   zeiα=(x+iy)(cosα+isinα) =(xcosα−ysinα)+i(ycosα+xsinα) Area =12|001xy1(xcosα−ysinα)(ycosα+xsinα)1| =12[xycosα+x2sinα−xycosα+y2sinα] =12sinα(x2+y2)=12|z|2sinα         [∵ |z|=(x2+y2)]