# Surface Area of Cylinders

(1) Right Circular Cylinder

1. Curved surface area = the perimeter x the height of the cylinder, i.e. $S = 2\pi rh$
2. The area of each of the flat surfaces, i.e. of the ends, $= \pi {r^2}$
3. 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

1. Curved surface area $= 2\pi Rh + 2\pi rh = 2\pi \left( {R + r} \right)h$
2. 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 semi-major axis and semi-minor axis and $h$ is the height, then

1. Volume $= \pi abh$
2. Curved surface area $= \pi \left( {a + b} \right)h$
3. Total surface area$= \pi \left( {a + b} \right)h + 2\pi ab$