Cylinder

Cylinder is a prism consisting of two parallel congruent circular bases.

In our daily life, objects like a piece of pipe, the drum of water filter etc. are the examples of the cylinder. The Cylinder has uniform circular cross sections. In the cylinder, there are two opposite parallel and congruent circular faces called the bases. The line segment CD joining the centers C and D of two circular bases of the cylinder are perpendicular to the base circle is called the axis of the cylinder. The length CD is called the height of the cylinder.

Surface area of a cylinder

As we see, there are two types of surfaces in the cylinder.
(i) Lateral (curved) surface
(ii) Plane ( circular base ) surfaces
Since cylinder is a prism, lateral surface area of prism is obtained by using the formula:
LSA = perimeter of the base× height or length of the prism. In case of cylinder,

\begin{align*} \text {curved surface area} &= \text {circumference of the base} \times \text {height of the cylinder.} \\&= 2 \pi r \times h \: square \: units \: or\: 2 \pi r \times l \text{square units}\\&=2 \pi rh \: or \: 2 \pi rl \: square \: units \end{align*}

Alternatively,

A hollow cylinder can be formed by rolling and joining two breadth of the rectangular sheet of paper as shown in the given figure.

Rectangular sheet of paper now change to the curved surface area of cylinder. The area of rectangle sheet of paper ABCD is \( l \times b.\) When rectangle is changed to cylinder, its length becomes the circumference of the base of the cylinder and its breadth becomes height 'h' of the cylinder.Therefore,the curved surface area of the cylinder\begin{align*}&= \text {(circumference of the base} \times \text {heigh of the cylinder) sq. units} \\ &= 2 \pi rh\\ \end{align*}

\(\boxed{\therefore CSA= 2 \pi rh \: or \: 2 \pi rl \: sq. \: unit}\)

Since at the base, there are two circles, so area of bases = 2πr² square units. Total surface area of the cylinder,

\begin{align*} \text{TSA} &= \text {curved surface area (CSA) + Area of bases}\\ or, \: TSA &= (2 \pi rh +2 \pi r^2 ) Sq. \: units \\ &= 2 \pi r ( r + h) \: Sq. \: units \\ \therefore TSA &= C(r + h) \text {where C is the circumference of the circle.}\\ \end{align*}

Volume of cylinder

Since a right circular cylinder is a prism, so the volume of prism is obtained as the product of base area and its height.

\begin{align*} \therefore Volume \: (V) &= Area \: of \ base \:circle \times height \\ &= \pi r^2 \times h \\ &=\pi r^2 h \: cubic \: units \\ \end{align*}

If diameter (d) is given,

\begin{align*} Volume \: (V) &= \pi \left ( \frac {d} {2} \right )^2 \times height \\ &= \frac {\pi d^2 h} {4} cubic \: unit\\ \end{align*}

Alternatively,

A right circular cylindrical shape is changed into the shape of cuboid as cylinder is cut into the even number of pieces ( as far as small pieces) and arranging them in the form of cuboid with length equal to half of the circumference of the base circle, breadth equal to the radius of base circle and height is equal to the height of the cylinder which is shown in the following figures:
[ Cut pieces of cylinder are arranged to form a cuboid.]

\begin{align*}\text {The length of cuboid } (l) = \frac {c} {2} = \pi r \:units \\ \text {The wide of cuboid } (b) &= r \: unit \\ \text {The height of cuboid } (h) &=h \: unit \\ \therefore \text {Volume of cuboid (V)} &= l \times b \times h \\ &= \pi r \times r \times h \text {cubic units}\\ &= \pi r^2h \: cubic \: units \\ \therefore \text {Volume of cylinder} &=\pi r^2h \: cubic \: units\\ \end{align*}