Surface Area of Cylinders
(1) Right Circular Cylinder

Curved surface area = the perimeter x the height of the cylinder, i.e. $$S = 2\pi rh$$

The area of each of the flat surfaces, i.e. of the ends, $$ = \pi {r^2}$$

The total surface area $$ = 2\pi rh + 2\pi {r^2} = 2\pi r\left( {r + h} \right)$$
Example:
Find the height of the solid circular cylinder if the total surface area is 600 sq.cm and the radius is 5cm.
Solution:
Here $$r = 5$$cm
Total surface area $$ = 2\pi {r^2} + 2\pi rh = 660$$
Or $$2\pi r\left( {r + h} \right) = 660$$
Or $$2 \times \frac{{22}}{7} \times 5\left( {5 + h} \right) = 660$$
Or $$5 + h = \frac{{660 \times 7}}{{5 \times 44}} = 21$$
$$\therefore $$ $$h = 16$$cm
Example:
A cylindrical vessel without a lid has to be coated on both its sides. If the radius of its base is $$\frac{1}{2}$$m and its height is 1.4m, calculate the cost of tin coating at the rate of $2.25 per 1000 sq.cm.
Solution:
Given that
Radius of the base of cylindrical vessel, $$r = \frac{1}{2}m = 50cm$$
Height, $$h = 1.4m = 140m$$
$$\therefore $$ Area to be tin coated $$ = 2\left( {{\text{curved surface + area of base}}} \right)$$
$$ = 2\left( {2\pi rh + \pi {r^2}} \right)$$
$$ = 2\pi r\left( {2h + r} \right)$$
$$ = 2 \times 3.14 \times 50\left( {2 \times 140 \times 50} \right) = 314 \times 330$$
$$ = 103620$$sq.cm
Now, the cost of tin coating per 1000sq.cm = $2.25
$$\therefore $$ Total cost of tin coating $$ = \frac{{2.25}}{{1000}} = 103620 = 233.15$$ dollars
(2) Hollow Circular Cylinder

Curved surface area $$ = 2\pi Rh + 2\pi rh = 2\pi \left( {R + r} \right)h$$

Total surface area $$ = 2\pi \left( {R + r} \right)h + 2\pi \left( {{R^2} – {r^2}} \right)$$
(3) Elliptic Cylinder
A cylinder with a base which is an ellipse is called an elliptic cylinder. If $$a$$ and $$b$$ are the semimajor axis and semiminor axis and $$h$$ is the height, then

Volume $$ = \pi abh$$

Curved surface area $$ = \pi \left( {a + b} \right)h$$

Total surface area$$ = \pi \left( {a + b} \right)h + 2\pi ab$$