Figure 9

Convergence of the two kinds of approximate values of the surface areas of the sphere. Let $S_{n^2}$ be a sum of divided areas of the 2D surface of a 3D solid. $S_{n^2}$ is expressed as </p><p></p><p> where two kinds of finite area element vectors $\overrightarrow{{\mathrm{\Delta }}^2{\sigma }_{j,k}}$ and $\overrightarrow{{\mathrm{\Delta }}^2{\sigma }_{k,j}}$ are divided areas of the 2D surface and $\overrightarrow{n_{j,k}}$ and $\overrightarrow{n_{k,j}}$ are corresponding unit normal vectors of respective divided area elements. As ${\sum }_{k=1}^n\overrightarrow{n_{k,k}}\cdot \overrightarrow{{\mathrm{\Delta }}^2{\sigma }_{k,k}}\to 0$, </p><p></p><p> coincides with $S_{n^2}$ at infinity $n\to \mathrm{\infty }$.