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 = 2πr^{2} 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*}
Note

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 h \\ &= \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*}
Curved surface area of the cylinder (CSA) = 2\(\pi\) rh
Total surface area of hollow cylinder = 2 \(\pi\)rh
Total surface area of lidless cylinder 2\(\pi\)rh + \(\pi\)r(2h + r)
Circular cylinder is a prism, so the volume of prism is obtained as the product of base area and its height, therefore
Volume = Area of base circle x height
\(\pi\) r\(^2\)h cubic units
Solution:
\begin{align*} \text{Curved surface area of cylinder} &= 2 \pi rh \\ or, 616 cm^2&= 2 \times \frac{22}{7} \times r \times 14 \: cm \\ or, 616cm^2 &= 88r \: cm \\ or, r&= \frac{616}{88}cm \\ \therefore radius (r) &= 7 \: cm \: _{Ans} \end{align*}
Solution:
Base radius (r) = 28 cm
Height (h) = 72 cm
Total surface area (T.S.A) = ?
We know that,
\begin{align*} T.S.A &= 2 \pi r (r+h)\\ &= 2\times \frac{22}{7} \times 28cm (28cm+72cm) \\ &= 8 \times 22cm (100cm)\\ &= 17600 \: sq. cm \: _{Ans} \end{align*}
Solution:
r = 14 cm, h = 13 cm
\begin{align*} Volume \: of \: cylinder &= \pi r^2 h \\ &= \frac{22}{7} \times (14cm)^2 \times 13cm \\ &= 8008cm^3 \: \: _{Ans}\end{align*}
Solution:
h = 60 cm
\(r=\frac{14}{2}=7cm\)
\begin{align*} \text{Volume of half part of a cylinder} &= \frac{1}{2} \times \pi r^2 h \\ &= \frac{1}{2} \times \frac{22}{7} \times 7^2 \times 60 \:cm^3 \\ &= 11\times 7 \times 60cm^3 \\ &= 4620 cm^3 \: \: _{Ans} \end{align*}
Solution:
Area of base \( (\pi r^2 ) = 154 cm^2 \)
height (h) = 14 cm
\begin{align*} Volume \: (v) &= \pi r^2 h \\ &= 154cm^2 \times 14 cm \\ &= 2156 \: cm^3 \: _{Ans} \end{align*}
Solution:
r + h = 10cm
\(Circumference = 2\pi r = 308 cm\)
\begin{align*}Total \: surface \: area &= 2 \pi r (r+h) \\ &= 308 \times 10 \\ &= 3080 cm^2 \: _{ans} \end{align*}
Solution:
\(r + h = 34 cm \\ Total \: surface \: area \: (S) = 2992cm^2 \\ By\: formula, \)
\begin{align*} S &= 2 \pi r (r+h)\\ or, 2992 &= 2 \times \frac{22}{7} \times r(34)\\ or, 2992 \times 7&= 1496r\\or, r &= \frac{2992\times 7}{1496}\\ \therefore r &= 14 \: cm \: _{Ans}\end{align*}
Solution:
Here,
\begin{align*} \text{surface area of cylindrical wood} &= 2 \pi rh \\ or, 308 &= 2 \pi h^2 \: [\because r = h ] \\ or, h^2 &= \frac{308}{2 \pi }\\ or, h^2 &= \frac{308}{2} \times \frac{22}{7}\\ or, h^2 &= \frac{308}{44} \times 7 \\ or, h &= \sqrt{49}\\ \therefore h &= 7 \: cm \: \: _{Ans}\end{align*}
Volume of the cylinder (V) = \(\pi\)r^{2}h
or, 1078 = \(\frac {22}7\)× r^{2}× 7
or, 1078 = 22r^{2}
or, r^{2} = \(\frac {1078}{22}\)
or, r^{2} = 49
or, r = \(\sqrt {49}\)
∴ r = 7 cm_{Ans}
Volume of cylinder (V) = \(\pi\)r^{2}h = area of base \(\times\) height
or, 1540 = 154 \(\times\) height
or, height = \(\frac {1540}{154}\)
∴ height of the solid = 10 cm_{Ans}
Radius of the base (r) = \(\frac {1.4}2\) = 0.7 m
height of the solid = ?
Volume of the solid (V) = 770 liters = \(\frac {770}{1000}\)m^{3} = 0.77 m^{3}
Nw,
Volume = \(\pi\)r^{2}h
or, 0.77 = \(\frac {22}{7}\) \(\times\) (0.7)^{2} \(\times\) h
or, 0.77 = \(\frac {22}7\) \(\times\) 0.49h
or, 0.77 = 22 \(\times\) 0.07h
or, 0.77 = 1.54 h
or, h = \(\frac {0.77}{1.54}\)
∴ height of the solid (h) = 0.5 m_{Ans}
The volume of a cylindrical can is 1.54 litre. If the area of its base is 77 m^{3}, find its height.
Volume (V) = 1.54 liters = 1.54 \(\times\) 1000 cm^{3} = 1540 cm^{3}
Area of base (A) = 77 cm^{2}
Height (h)= ?
Here,
V = A \(\times\) h
or, 1540 = 77 \(\times\) h
or, h = \(\frac {1540}{77}\)
∴ height of the can (h) = 20 cm_{Ans}
Here,
r = 6 cm
By Question,
curved surface area = \(\frac 23\) \(\times\) total surface area
or, 2\(\pi\)rh = \(\frac 23\) \(\times\) 2\(\pi\)r (r + h)
or, h = \(\frac 23\) (r + h)
or, 3h = 2(6 + h)
or, 3h = 12 + 2h
or, 3h  2h = 12
∴ height of the cylinder (h) = 12 cm_{Ans}
Here,
h = 25 cm
r_{1} = 4 cm
t = 1 cm
r_{2} = (4  1)cm = 3 cm
Now,
\begin{align*} \text {Volume of the metal (V)} &= \pi(r_1^2  r_2^2)h\\ &=\frac {22}7(4^2  3^2) \times 25\\ &= \frac {22}{7} \times 7 \times 25\\ &= 550 cm^3_{Ans}\\ \end{align*}
Base area of a cylinder (A) = 154 cm^{2}
Curved Surface Area (CSA) = 880 cm^{2}
Total Surface Area (TSA) = ?
By formula,
\begin{align*} TSA &= 2A + CSA\\ &= (2 \times 154 cm^2) + 880 cm^2\\ &= (308 + 880)cm^2\\ &= 1188cm^2_{Ans}\\ \end{align*}

The area of the base of a cylindrical chimney of 12 m high is 616 square metres, Find the volume of the chimney.
7320 cm^{3}
7291 cm^{3}
7393 cm^{3}
7319 cm^{3}

The volume of a cylinder solid is 1320 m^{3}. If the area of its base is 264 m calculate the height of the cylindrical solid.
3 m
6 m
5 m
7 m

The volume of a cylindrical can is 1.54 litre. If the area of its base is 77 m^{2}. Find its height.
19 cm
20 cm
15 cm
12 cm

A cylindrical water tank contains 462000 litres of water. If tits radius is 3.5 m, what is its height?
15 cm
12 cm
17 cm
8 cm

The volume of a cylinder is 770 cm^{3}. If the height of the cylinder is 5 cm, find the radius of the base of the cylinder.
5 cm
10 cm
8 cm
7 cm

The base radius and the height of a cylinder are in the ratio 5:7.If the volume of the cylinder is 550 cubic cm, find the radius of the base of the cylinder.
5 cm
6 cm
3 cm
7 cm

The base radius and the height of a cylinder are in the ratio 6:7. If the volume of the cylinder is 6336 cubic cm, find the radius of the base of the cylinder.
12 cm
13 cm
14 cm
15 cm

The diameter of a cylindrical log of wood is 42 cm. If the area of its curved surface is 11880 sq.cm, find its height.
75 cm
60 cm
90 cm
70 cm

The diameter of the cylindrical log wood is 28 cm.If the area of the curved surface is 7392 sq. cm, find its height.
60 cm
70 cm
84 cm
64 cm

If the height and radius of a cylindrical wood are equal and curved surface area is 308 cm^{ 2}.Find the height of the wood.
5 cm
10 cm
7 cm
9 cm

The curved surface area and height of a cylinder are 616 cm^{2} and 14 cm respectively.Find the radius of the cylinder,
9 cm
10 cm
7 cm
8 cm

The radius and height of a cylinder are equal.If its curved surface area is 1232 cm^{2}, find the volume of the cylinder.
8628 cm^{3}
8623 cm^{3}
8724 cm^{3}
8624 cm^{3}

The sum of the height and radius of a cylinder is 32cm. If the area of total surface is 1408 cm^{2}, find the volume and perimeter of the base of the cylinder.
47 cm,3555 cm^{3}
44 cm,3850 cm^{3}
45 cm, 3550 cm^{3}
46 cm,3680 cm^{3}

The curved surface area of a solid cylinder is equal to 3/4 of its total surface area.find the radius and height of it, if its total surface area is 1232 cm^{2}.
7 cm, 21 cm
3 cm,12 cm
8 cm,25 cm
5 cm, 20 cm

The height of a right circular cylinder is 14 cm. If two times the sum of the area of its two circular faces is equal to its curved surface area, find the radius of the base.
8 cm
7 cm
10 cm
15 cm

You scored /15
Any Questions on Cylinder ?
Please Wait...
Discussions about this note
Forum  Time  Replies  Report 

cylinderIf the sum of height and radius of base of right cylinder is 34 cm and its total surface area is 2992 cm squared. Find its diameter 
Jan 11, 2017 
0 Replies View Replies 