# Volume and surface area of cylinder

In this chapter, we will discuss the concept of right circular cylinder and learn the formula to calculate the volume, curved and total surface area.

## What is cylinder ?

The cylinder is a 3 dimensional shape, consist of two circles of same radius parallel to each other joined by the curved surface in between.

Note that the center of the parallel circles lie in the same line so that the cylinder remains perfectly vertical.

Given above is the figure of right circular cylinder.

The above cylinder is formed by two circles of radius ” r” kept at a distance ” h ” from each other. These two circles are joined by curved surface to form cylinder.

Note that the line AB is called radius of cylinder and line AO is called axis of cylinder.

### Parts of Right circular cylinder

Base
The circle at the bottom is called base of the cylinder

Top Base
The circle at the top is called top base

Axis of cylinder
The imaginary line joining two base is called axis of cylinder

The radius of base is referred as radius of cylinder. In right circular cylinder, radius of bottom and top circle is the same.

Lateral Surface
The curved surface between the two circles is called lateral surface of cylinder.

### Properties of cylinder

Given below are some properties of cylinder;

(a) The top and bottom of cylinder are parallel and congruent to each other

(b) In right circular cylinder, the axis is perpendicular to the base.

(c) There are mainly four types of cylinder;

(i) Oblique cylinder
Here the axis of the cylinder is not perpendicular to its base. The shape of this cylinder looks like a tilted building.

(ii) Elliptic cylinder
The cylinder in which the base is in the form of ellipse is called elliptic cylinder

(iii) Cylindrical shell
It’s a cylinder inside a cylinder.

(iv) Right circular cylinder
Its a cylinder with circular base with axis perpendicular to the base.

Here we will discuss right circular cylinder in detail. I hope you understood the basic concepts, let’s move to understand different formulas.

### Formulas for right circular cylinder

#### Volume of right circular cylinder

The space enclosed by the cylinder is called volume of cylinder.

The formula is given as;

\mathtt{Volume\ of\ cylinder\ =\ \pi \ r^{2} \ h}

We get volume in cubic units ( \mathtt{cm^{3} ,\ m^{3}} )

#### Total surface area of cylinder

The total surface area of cylinder includes the area of two circular bases and the curved surface area in between.

Total Surface area = Base + Top base + curved surface

The formula for total surface area is given as;

TSA = 2𝜋r ( h + r )

We get the surface area in square units ( \mathtt{cm^{2} ,\ m^{2}} )

#### Lateral surface area of cylinder

The area of the curved part of the cylinder is called lateral surface area.

The formula for curved surface area is given as;

Curved surface area = circumference x height

Curved surface area = 2 𝜋 r h

We get the lateral surface are in square units ( \mathtt{cm^{2} ,\ m^{2}} )

You need to memorize all the above formulas as they help us to solve questions faster. Given below are collection of questions related to right circular cylinder.

## Questions on right circular cylinder

Question 01
Calculate the volume of cylinder with radius 7 cm and height 10 cm.

Solution
r = 7 cm
h = 10 cm

\mathtt{Volume\ =\ \pi r^{2} h}\\\ \\ \mathtt{Volume\ =\ \frac{22}{7} \times 7\times 7\times 10\ =1540\ cm^{2}}

Hence, 1540 sq. cm is the required volume.

Question 02
Find the lateral surface are of cylinder with diameter 6 meter and height 5 meter.

Solution
diameter = 6 meter
radius = 3 meter

height = 5 meter

Curved surface area = 2𝜋rh

Curved surface area = \mathtt{\ 2\times \frac{22}{7} \times 3\times 5\ =94.3\ m^{2}}

Hence, 94.3 sq meter is the required curved surface area.

Question 03
A factory owner decide to paint the outer wall of circular chimney of radius 0.5 meter and height 10 meter. If the cost of painting is 2$per sq. meter, calculate the cost to paint the whole chimney. Solution r = 0.5 meter h = 10 meter Here we have to calculate the curved surface area of cylinder to find the exact area which is to be painted. Curved surface area = 2𝜋rh Curved surface area = \mathtt{2\times \frac{22}{7} \times 0.5\times 10\ =31.43\ m^{2}} So, 31.43 sq. meter of area is to be painted. Now let’s calculate the cost. 1 sq. meter ⟹ 2$

31.43 sq meter ⟹ 31.43 x 2 = 62.86\$

Hence, the total cost to paint will be 62.86 dollars.

Question 04
Calculate the ratio of curved to total surface area of cylinder with height 15 cm and radius 4 cm.

Solution
Curved surface area formula = 2𝜋rh

Total surface area = 2𝜋r(h + r)

curved area/total area = \mathtt{\frac{2\pi rh}{2\pi r( h+r)} \ =\ \frac{h}{h+r}}

Putting the values;

curved/total area = \mathtt{\frac{15}{15+4} =\frac{15}{19}}

Hence, 15 : 19 is the required ratio

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