Subject: Compulsory Mathematics
A hollow cylinder can be formed by rolling and joining two breadth of the rectangular sheet of paper. A Cylinder is a prism consisting of two parallel congruent circular bases.
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.
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*}
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

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*}
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:
Here,
\(radius\: (r) = \frac{8cm}{2} = 4cm\)
\begin{align*} \text{Curved surface area of cylinder } &= 2 \pi rh \\ &= 2 \times \frac{22}{7} \times 4 \times 140 cm^2 \\ &= 3520 cm^2 \: _{Ans} \end{align*}
The curved surface area and height of a cylinder are 616 sq. cm and 14 cm respectively. Find the radius of the cylinder.
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*}
Find the total surface area of a cylinder whose base radius is 28 cm and height is 72 cm.
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*}
A cylinder has a radius of the base 14 cm and height 13 cm. find the volume of a cylinder. \(\pi =\frac{22}{7}\)
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*}
A 60 cm high cylinder with 14cm its diameter cut vertically into two equal halves. What is the volume of a half part?
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*}
The length and base area of cylinder given in the figure alongside are 14 cm and 154 sq. cm respectively. Find the volume of the cylinder.
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*}
If sum of radius and height of cylinder is 10 cm , and circumference of base is 308 cm, find the total surface area of that cylinder.
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*}
The sum of the height and the radius of the base of a cylinder is 34 cm. If the total surface area of the cylinder is 2992 square cm, find the radius of its base.
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*}
If height and radius of a cylindrical wood are equal and curved surface area is 308 cm^{2} , find height of it.
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*}
The volume of a cylinder is 1078 cm^{3} and height is 7 cm. Find the radius of the base.
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}
A cylindrical solid object has a volume of 1540 cubic cm. If its base area is 154 square cm; what will be its height?
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}
A cylindrical water vessel of diameter 1.4 meter holds 770 liters of water. Calculate the height of the vessel. (1000 liters = 1 m^{3} and \(\pi\) = \(\frac {22}7\))
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}
If the curved surface area of a cylinder is two third of the total surface area and radius of its base is 6 cm, find the height of the cylinder.
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}
The radius of 25 cm long cylindrical metal is 4 cm. If the thickness of the metal is 1 cm, find the volume of the metal.
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*}
The base area of a cylinder is 154 sq. cm and the curved surface area is 880 sq. cm. What is the total surface area? Find it.
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*}
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