91Ó°ÊÓ

Let’s choose a cylindrical geometric shape. Its total surface area is the sum of the curved surface of the cylinder and the area of the top and bottom circular surfaces.

The expression$b(c−d)−e$ can be written as

$\begin{array}{l}b(c−d)÷e=\left[\right(b×c)−(b×d\left)\right]÷e\\ \text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}=(bc−bd)÷e\\ \text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}=\frac{(bc−bd)}{e}\end{array}$

It is known that the area of a circle is $\mathrm{Ï€}{r}^2$. A cylinder has a circular top and a circular bottom. Let the radius of the cylinder be r.Then the total surface area of the two circular regions is $(\mathrm{Ï€}{r}^2+\mathrm{Ï€}{r}^2)=2\mathrm{Ï€}{r}^2$.

Again, it is further known that the curved surface area of a cylinder with radius r and height h is given by $2\mathrm{Ï€}rh$.

Thus, the total surface area of the cylinder is the sum of the curved surface area and the area of the two circular regions which is $2\mathrm{Ï€}rh+2\mathrm{Ï€}{r}^2$.

Hence, the geometric formula for the total surface area of a cylinder with radius r and height h is $2\mathrm{Ï€}rh+2\mathrm{Ï€}{r}^2$.