Question

Class 10

Mathematics

Chapter - Surface area and volume .

Formulas Needed

Quality Answer Needed​

Answer 1

$\sf{ \orange { \underline { \underline { Surface \: Area \: And \: Volume \: - }}}}$

For the chapter , Surface Area And Volume , all the formulas and all the various types of questions you can encounter are listed here !

First , let me explain what this chapter exactly deals with .

For any 2 dimensional or 3 dimensional object , it occupies a certain space and has a certain volume .We are basically trying to calculate these for each of the objects .

• The surface area of any given object refers to the area occupied by the surface of the object.

• The volume is the amount of space available within the object.

Here , we will be dealing with the following figures -

All 3 dimensional Objects like

( a ) Cubiod

( b ) Cube

( c ) Cylinder

( d ) Cone [ And it's frustum ]

( e ) Sphere / Hemisphere

( f ) Spherical Shell etc

For three dimensions objects , there are two types of surface areas possible -

• Lateral / curved Surface Area - Lateral surface area refers to the area of all the sides of the object excluding the bottom and top , if it exists

• Total surface Area - Total surface area refers to the total area covered by all the sides of an object .

Let us start three two dimensional figures first -

1. Cube

$\setlength{\unitlength}{4mm}\begin{picture}(10,6)\thicklines\put(0,1){\line(0,1){10}}\put(0,1){\line(1,0){10}}\put(10,1){\line(0,1){10}}\put(0,11){\line(1,0){10}}\put(0,11){\line(1,1){5}}\put(10,11){\line(1,1){5}}\put(10,1){\line(1,1){5}}\put(0,1){\line(1,1){5}}\put(5,6){\line(1,0){10}}\put(5,6){\line(0,1){10}}\put(5,16){\line(1,0){10}}\put(15,6){\line(0,1){10}}\put(4.6,-0.5){\bf\large a m}\put(13.5,3){\bf\large a m}\put(-4,5.8){\bf\large a m}\end{picture}$

A cube has all the sides equal .

Let the side of the cube be a units .

Lateral surface area - 4 a² .

Total surface Area - 6 a².

Volume - a³

2. Cubiod -

For a cuboid , having the length , breadth and height as x , y and z respectively ;

$\setlength{\unitlength}{0.74 cm}\begin{picture}\thicklines\put(5.6,5.4){\bf A}\put(11.1,5.4){\bf B}\put(11.2,9){\bf C}\put(5.3,8.6){\bf D}\put(3.3,10.2){\bf E}\put(3.3,7){\bf F}\put(9.25,10.35){\bf H}\put(9.35,7.35){\bf G}\put(3.5,6.1){\sf x\:cm}\put(7.7,6.3){\sf y\:cm}\put(11.3,7.45){\sf z\:cm}\put(6,6){\line(1,0){5}}\put(6,9){\line(1,0){5}}\put(11,9){\line(0,-1){3}}\put(6,6){\line(0,1){3}}\put(4,7.3){\line(1,0){5}}\put(4,10.3){\line(1,0){5}}\put(9,10.3){\line(0,-1){3}}\put(4,7.3){\line(0,1){3}}\put(6,6){\line(-3,2){2}}\put(6,9){\line(-3,2){2}}\put(11,9){\line(-3,2){2}}\put(11,6){\line(-3,2){2}}\end{picture}$

Lateral Surface Area - 2( l + b ) h

Total surface area - 2( lb + bh + lh )

Volume - lbh

Diagonal - √{ l² + b² + h² }

3. Cylinders ( Right circular Cylinder )

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\thicklines\multiput(-0.5,-1)(26,0){2}{\line(0,1){40}}\multiput(12.5,-1)(0,3.2){13}{\line(0,1){1.6}}\multiput(12.5,-1)(0,40){2}{\multiput(0,0)(2,0){7}{\line(1,0){1}}}\multiput(0,0)(0,40){2}{\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\multiput(18,2)(0,32){2}{\sf{r}}\put(9,17.5){\sf{h}}\end{picture} </p><p>$

Lateral Surface Area : 2πrh

Total Surface Area : 2πr(h+r)

Volume : πr²h

4. Cylinders ( Hollow cylinder)

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(4,0){2}{\line(0,1){6}}\multiput(1,0)(2,0){2}{\line(0,1){6}}\multiput(0,0)(0,6){2}{\qbezier(0,0)(2,-0.6)(4,0)}\multiput(0,0)(0,6){2}{\qbezier(0,0)(2,0.6)(4,0)}\multiput(1,0)(0,6){2}{\qbezier(0,0)(1,-0.2)(2,0)}\multiput(1,0)(0,6){2}{\qbezier(0,0)(1,0.2)(2,0)}\multiput(2,0.07)(0,0.3){20}{\line(0,1){0.2}}\multiput(2,4)(0.3,0){7}{\line(1,0){0.2}}\multiput(2,2)(-0.27,0){4}{\line(-1,0){0.2}}\put(1.4,1.5){\bf\large r}\put(3.35,3.45){\bf R}\put(1.4,3){\bf H}\end{picture}$

External CSA - 2πRh

Internal CSA- 2πrh

Total surface area : 2πrh + 2πRh + 2πR² - 2πr²

Volume : πR²h - πr²h

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For continuation -

https://brainly.in/question/28918412

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Answer 2

$\Large{\underline {\gray{\mathrm{ Surface \:Area\:and\:Volume \::}}}}$

In this chapter :

❍ We will come across many 3D shapes are :

• Cuboid
• Cube
• Cylinder
• Cone
• Sphere
• Hemisphere

⠀⠀⠀⠀Now , Here we will get to know some formulas of some 3D Shapes :

$\large{\bf{Cuboid \::}}$

• $\:\bf{Volume \:of\:Cuboid \:= \:Length \times Breadth \times Height \:\:cu.units}$
• $\:\bf{Total \:Surface \:Area\:of\:Cuboid \:= \:\small{2(Length \times Breadth + Breadth \times Height + Length \times Height) \:} \:\:sq.units}$
• $\:\bf{Lateral \:Surface \:Area\:of\:Cuboid \:= \:2( Length + Breadth) Height \:\:sq.units}$
• $\:\bf{Diagonal \:of\:Cuboid \:= \:\sqrt {Length^2 + Breadth^2 +Height^2 }\:\:units}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large{\bf{Cube \::}}$

• $\:\bf{Volume \:of\:Cuboid \:= \:Edge \times Edge \times Edge \:\:cu.units}$
• $\:\bf{Total \:Surface \:Area\:of\:Cube \:= 6 \times Edge^2\:\:sq.units}$
• $\:\bf{Lateral \:Surface \:Area\:of\:Cuboid \:= \: 4 \times Edge ^2\:\:sq.units}$
• $\:\bf{Diagonal \:of\:Cuboid \:= \:\sqrt {3 } \times Edge \:\:units}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large{\bf{Cylinder \::}}$

• $\:\bf{Volume \:of\:Cylinder \:=\pi \times Radius ^2 \times Height \:\:p \:\:cu.units}$
• $\:\bf{Total \:Surface \:Area\:of\:Cylinder \:= 2 \times \pi \times Radius ( Radius + Height)\:\:sq.units}$
• $\:\bf{Curved \:Surface \:Area\:of\:Cylinder \:=\:2 \times \pi\times Radius \times Height \:\:sq.units \:}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large{\bf{Cone \::}}$

• $\:\bf{Volume \:of\:Cone \:=\dfrac{1}{3}\times \pi \times Radius ^2 \times Height \:\:p \:\:cu.units}$
• $\:\bf{Total \:Surface \:Area\:of\:Cone \:= \pi \times Radius ( Radius + Length)\:\:sq.units}$
• $\:\bf{Curved \:Surface \:Area\:of\:Cone \:=\: \pi\times Radius \times Length \:\:sq.units \:}$
• $\:\bf{Slant \:Height \:of\:Cone \:= \:\sqrt {Radius ^2 + Height^2 }\:\:units}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large{\bf{Sphere \::}}$

• $\:\bf{Total \:Surface \:Area\:of\:Sphere \:= 4 \times \pi \times Radius^2 \:\:sq.units}$
• $\:\bf{Volume \:of\:Sphere \:=\dfrac{4}{3}\times \pi \times Radius ^3 \:\: \:\:cu.units}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large{\bf{Hemisphere \::}}$

• $\:\bf{Volume \:of\:Hemisphere \:=\dfrac{2}{3}\times \pi \times Radius ^3 \:\: \:\:cu.units}$
• $\:\bf{Curved \:Surface \:Area\:of\:Hemisphere \:=\:2 \times \pi\times Radius^2 \:\:sq.units \:}$
• $\:\bf{Total \:Surface \:Area\:of\:Hemisphere \:= 3 \times \pi \times Radius^2 \:\:sq.units}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large{\bf{Frustum \::}}$

• $\:\bf{Volume \:of\:Hemisphere \:=\tiny{\dfrac{1}{3}\times \pi \times Height ( Radius _{1}\:^2 + Radius_{2}\:^2 + Radius _{1} + Radius _{2})\:}\: \:\:cu.units}$
• $\:\bf{Total \:Surface \:Area\:of\:Hemisphere \:= \tiny{ \pi \times Radius_{1} \:^2 + \pi \times Radius_{2}\:^2+ \pi \times Radius_{1}( Length + Radius_{1})}\:\:sq.units}$
• $\:\bf{Curved \:Surface \:Area\:of\:Hemisphere \:=\: \pi\times (Radius_{1} + Radius_{2})Length \:\:sq.units \:}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀