Surface area of a cylinder

Surface area of a cylinder

Any cylindrical object has the form of the following figure:

The parts of the figure are the top, base and curved surface. The top and base have the same diameter.

When a cylinder is opened along its height, the resulting surfaces are shown in the following figure:

When a cylinder is cut along its height, it results into three shapes which are the two circles and a rectangle. In order to obtain the surface area of a cylinder, add areas of the circles and the area of the rectangle. The formula for finding the surface area of a cylinder is obtained as follows:

The circumference of a circle is the same as the width of the rectangle obtained after opening the cylinder.

$= \pi dh + \frac{2\pi d^2}{4}$

Using the radius of the cylinder, the formula is given by

Surface area of a cylinder $= 2\pi rh + 2\pi r^2$

$= 2\pi r(r + h)$

Therefore, the surface area of a cylinder $= 2\pi r(r + h)$.

Example 1

Find the surface area of the following cylinder:

Solution

By using the formula involving a diameter

Surface area of a cylinder $= \pi dh + \frac{2\pi d^2}{4}$

$= \frac{22}{7} \times 21\text{ cm} \times 50\text{ cm} + 2 \times \frac{22}{7} \times 21\text{ cm} \times \frac{21\text{ cm}}{4}$

$= 3\,300\text{ cm}^2 + 693\text{ cm}^2$

$= 3\,993\text{ cm}^2$

Therefore, the surface area of a cylinder is $3\,993\text{ cm}^2$

Example 2

A closed empty bucket is cut at its top as shown in the following figure. Find the surface area of the bucket if its height is 110 cm, diameter 84 cm and the diameter of the hole on the top is 28 cm.

Solution

Surface area of a bucket $=$ surface area of a whole figure $-$ area of the hole

$= \frac{22}{7} \times 84\text{ cm} \times 110\text{ cm} + 2 \times \frac{22}{7} \times 84\text{ cm} \times \frac{84\text{ cm}}{4} - \frac{22}{7} \times 28\text{ cm} \times \frac{28\text{ cm}}{4}$

$= 29\,040\text{ cm}^2 + 11\,088\text{ cm}^2 - 616\text{ cm}^2$

$= 39\,512\text{ cm}^2$

Therefore, the surface area of the bucket is $39\,512\text{ cm}^2$