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Note on Cylinder

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

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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.

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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*}

 

 

Note

  1. TSA of hollow cylinder =2πrh (A cylinder whose two ends are opened )
  2. TSA of lidless cylinder ( A cylinder whose one side is opened ) 2πrh +πr2=πr ( 2h + r )

 

Volume of cylinder

 

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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,

 

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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

 

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Very Short Questions

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*}

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) = \(\pi\)r2h

or, 1078 = \(\frac {22}7\)× r2× 7

or, 1078 = 22r2

or, r2 = \(\frac {1078}{22}\)

or, r2 = 49

or, r = \(\sqrt {49}\)

∴ r = 7 cmAns

Volume of cylinder (V) = \(\pi\)r2h = area of base \(\times\) height

or, 1540 = 154 \(\times\) height

or, height = \(\frac {1540}{154}\)

∴ height of the solid = 10 cmAns

Radius of the base (r) = \(\frac {1.4}2\) = 0.7 m

height of the solid = ?

Volume of the solid (V) = 770 liters = \(\frac {770}{1000}\)m3 = 0.77 m3

Nw,

Volume = \(\pi\)r2h

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 mAns

Volume (V) = 1.54 liters = 1.54 \(\times\) 1000 cm3 = 1540 cm3

Area of base (A) = 77 cm2

Height (h)= ?

Here,

V = A \(\times\) h

or, 1540 = 77 \(\times\) h

or, h = \(\frac {1540}{77}\)

∴ height of the can (h) = 20 cmAns

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 cmAns

Here,

h = 25 cm

r1 = 4 cm

t = 1 cm

r2 = (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 cm2

Curved Surface Area (CSA) = 880 cm2

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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If 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


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