06 - Let's Recap - Surface Area and Volume - Class 10 - Maths Class 10 Notes | EduRev

Crash Course for Class 10 Maths by Let's tute

Class 10 : 06 - Let's Recap - Surface Area and Volume - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


 
 
SURFACE AREAS AND VOLUMES
 
Solid Figures: The objects which occupy space 
width, depth and height.) are called solids. The
Example: Sphere, cone, cube, 
Surface Area: The sum of the areas of the plane figures making up the boundary of a solid 
figure is called its surface area
 
Volume: The measure of part of space occupied by a solid is called its 
 
Lateral Surface Area: Lateral surface area is the area of all the sides of an object where we 
don’t have to count the areas of
i.e. Lateral area = Surface area 
 
Cuboid: A cuboid is a box-shaped solid object
angles. All of its faces are rectangles.
• Lateral Surface Area
• Total Surface Area = 
• Volume of cuboid = 
                  Where, l = length 
                              b = breadth
                              h = height 
 
Cube: If the faces of a rectangular parallelepiped be squares, then it is called 
• Lateral Surface Area = 
• Total Surface Area = 
• Volume of cube = 
                   Where, a = side.                                     
 
 
 
 
Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its 
sides is called a right circular cylinder.
• Curved Surface Area = 
• Total Surface Area = 
• Volume = r
2
p
                   Where, r = radius of
                               h = height 
 
 
SURFACE AREAS AND VOLUMES 
The objects which occupy space (i.e. they are 3 – dimensional figures having 
are called solids. The solid figures are derived from the plane figures. 
, cylinder, rectangular prism, etc. 
 
The sum of the areas of the plane figures making up the boundary of a solid 
surface area. 
: The measure of part of space occupied by a solid is called its volume
: Lateral surface area is the area of all the sides of an object where we 
don’t have to count the areas of the top and the bottom. 
Lateral area = Surface area – Area of two ends 
shaped solid object. It has six flat sides and all the angles are right 
ll of its faces are rectangles. 
Lateral Surface Area = h b l ) ( 2 + 
Total Surface Area = ) ( 2 hl bh lb + + 
Volume of cuboid = h b l × × 
 
b = breadth 
 
If the faces of a rectangular parallelepiped be squares, then it is called 
Lateral Surface Area = 
2
4a 
Total Surface Area = 
2
6a 
Volume of cube = 
3
) (a 
                                                    
A solid obtained by revolving a rectangular lamina about one of its 
called a right circular cylinder. 
Curved Surface Area = rh p 2 
Total Surface Area = ) ( 2 h r r + p 
h
2
 
 
, r = radius of circular base. 
 
dimensional figures having 
are derived from the plane figures.  
The sum of the areas of the plane figures making up the boundary of a solid 
volume. 
: Lateral surface area is the area of all the sides of an object where we 
It has six flat sides and all the angles are right 
 
If the faces of a rectangular parallelepiped be squares, then it is called cube. 
A solid obtained by revolving a rectangular lamina about one of its 
Page 2


 
 
SURFACE AREAS AND VOLUMES
 
Solid Figures: The objects which occupy space 
width, depth and height.) are called solids. The
Example: Sphere, cone, cube, 
Surface Area: The sum of the areas of the plane figures making up the boundary of a solid 
figure is called its surface area
 
Volume: The measure of part of space occupied by a solid is called its 
 
Lateral Surface Area: Lateral surface area is the area of all the sides of an object where we 
don’t have to count the areas of
i.e. Lateral area = Surface area 
 
Cuboid: A cuboid is a box-shaped solid object
angles. All of its faces are rectangles.
• Lateral Surface Area
• Total Surface Area = 
• Volume of cuboid = 
                  Where, l = length 
                              b = breadth
                              h = height 
 
Cube: If the faces of a rectangular parallelepiped be squares, then it is called 
• Lateral Surface Area = 
• Total Surface Area = 
• Volume of cube = 
                   Where, a = side.                                     
 
 
 
 
Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its 
sides is called a right circular cylinder.
• Curved Surface Area = 
• Total Surface Area = 
• Volume = r
2
p
                   Where, r = radius of
                               h = height 
 
 
SURFACE AREAS AND VOLUMES 
The objects which occupy space (i.e. they are 3 – dimensional figures having 
are called solids. The solid figures are derived from the plane figures. 
, cylinder, rectangular prism, etc. 
 
The sum of the areas of the plane figures making up the boundary of a solid 
surface area. 
: The measure of part of space occupied by a solid is called its volume
: Lateral surface area is the area of all the sides of an object where we 
don’t have to count the areas of the top and the bottom. 
Lateral area = Surface area – Area of two ends 
shaped solid object. It has six flat sides and all the angles are right 
ll of its faces are rectangles. 
Lateral Surface Area = h b l ) ( 2 + 
Total Surface Area = ) ( 2 hl bh lb + + 
Volume of cuboid = h b l × × 
 
b = breadth 
 
If the faces of a rectangular parallelepiped be squares, then it is called 
Lateral Surface Area = 
2
4a 
Total Surface Area = 
2
6a 
Volume of cube = 
3
) (a 
                                                    
A solid obtained by revolving a rectangular lamina about one of its 
called a right circular cylinder. 
Curved Surface Area = rh p 2 
Total Surface Area = ) ( 2 h r r + p 
h
2
 
 
, r = radius of circular base. 
 
dimensional figures having 
are derived from the plane figures.  
The sum of the areas of the plane figures making up the boundary of a solid 
volume. 
: Lateral surface area is the area of all the sides of an object where we 
It has six flat sides and all the angles are right 
 
If the faces of a rectangular parallelepiped be squares, then it is called cube. 
A solid obtained by revolving a rectangular lamina about one of its 
 
 
 SURFACE AREAS AND VOLUME
 
 
Right Circular Cone: A right circular cone is a solid generated by the revolution of a right 
angled triangle about one of its side
• Curved Surface Area
• Total Surface Area = 
• Volume = 
3
1
p
                   Where, r = radius of base.
                               h = height 
                               l = slant height
 
Sphere: A sphere is a solid generated by the revolution of a semi
 
• Curved Surface Area = 
• Total Surface Area = 
• Volume = 
3
4
p
                    Where, r = radius
 
Spherical shell: A spherical shell is a generalization of an annulus to three dimensions
• Volume =
3
4
p
                  Where, 
1
r and 
2
r are its 
• Total Surface Area
 
Hemisphere: A plane passing through the centre
each called a hemisphere.    
• Curved Surface Area = 
• Total Surface Area = 
• Volume = 
3
2
p
                   Where, r = radius.
 
 
 
 
 
 
SURFACE AREAS AND VOLUMES
right circular cone is a solid generated by the revolution of a right 
angled triangle about one of its side containing the right angle as axis.                    
 
Curved Surface Area = rl p 
=
2 2
h r r + p 
Total Surface Area = ) ( r l r + p 
h r
2
p 
, r = radius of base. 
 
l = slant height 
sphere is a solid generated by the revolution of a semi-circle about its diameter.
Curved Surface Area = 
2
4 r p                                                     
Total Surface Area = 
2
4 r p 
3
r p 
, r = radius 
A spherical shell is a generalization of an annulus to three dimensions
) (
3
2
3
1
r r - 
are its external and internal radii respectively.
Total Surface Area = ) ( 4
2
2
2
1
r r + p 
A plane passing through the centre of a sphere cuts the sphere in two equal parts 
 
Curved Surface Area = 
2
2 r p                                                     
Total Surface Area = 
2
3 r p 
3
r p 
, r = radius. 
S 
right circular cone is a solid generated by the revolution of a right 
                     
circle about its diameter. 
                                                      
A spherical shell is a generalization of an annulus to three dimensions. 
respectively. 
cuts the sphere in two equal parts 
                                                      
Page 3


 
 
SURFACE AREAS AND VOLUMES
 
Solid Figures: The objects which occupy space 
width, depth and height.) are called solids. The
Example: Sphere, cone, cube, 
Surface Area: The sum of the areas of the plane figures making up the boundary of a solid 
figure is called its surface area
 
Volume: The measure of part of space occupied by a solid is called its 
 
Lateral Surface Area: Lateral surface area is the area of all the sides of an object where we 
don’t have to count the areas of
i.e. Lateral area = Surface area 
 
Cuboid: A cuboid is a box-shaped solid object
angles. All of its faces are rectangles.
• Lateral Surface Area
• Total Surface Area = 
• Volume of cuboid = 
                  Where, l = length 
                              b = breadth
                              h = height 
 
Cube: If the faces of a rectangular parallelepiped be squares, then it is called 
• Lateral Surface Area = 
• Total Surface Area = 
• Volume of cube = 
                   Where, a = side.                                     
 
 
 
 
Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its 
sides is called a right circular cylinder.
• Curved Surface Area = 
• Total Surface Area = 
• Volume = r
2
p
                   Where, r = radius of
                               h = height 
 
 
SURFACE AREAS AND VOLUMES 
The objects which occupy space (i.e. they are 3 – dimensional figures having 
are called solids. The solid figures are derived from the plane figures. 
, cylinder, rectangular prism, etc. 
 
The sum of the areas of the plane figures making up the boundary of a solid 
surface area. 
: The measure of part of space occupied by a solid is called its volume
: Lateral surface area is the area of all the sides of an object where we 
don’t have to count the areas of the top and the bottom. 
Lateral area = Surface area – Area of two ends 
shaped solid object. It has six flat sides and all the angles are right 
ll of its faces are rectangles. 
Lateral Surface Area = h b l ) ( 2 + 
Total Surface Area = ) ( 2 hl bh lb + + 
Volume of cuboid = h b l × × 
 
b = breadth 
 
If the faces of a rectangular parallelepiped be squares, then it is called 
Lateral Surface Area = 
2
4a 
Total Surface Area = 
2
6a 
Volume of cube = 
3
) (a 
                                                    
A solid obtained by revolving a rectangular lamina about one of its 
called a right circular cylinder. 
Curved Surface Area = rh p 2 
Total Surface Area = ) ( 2 h r r + p 
h
2
 
 
, r = radius of circular base. 
 
dimensional figures having 
are derived from the plane figures.  
The sum of the areas of the plane figures making up the boundary of a solid 
volume. 
: Lateral surface area is the area of all the sides of an object where we 
It has six flat sides and all the angles are right 
 
If the faces of a rectangular parallelepiped be squares, then it is called cube. 
A solid obtained by revolving a rectangular lamina about one of its 
 
 
 SURFACE AREAS AND VOLUME
 
 
Right Circular Cone: A right circular cone is a solid generated by the revolution of a right 
angled triangle about one of its side
• Curved Surface Area
• Total Surface Area = 
• Volume = 
3
1
p
                   Where, r = radius of base.
                               h = height 
                               l = slant height
 
Sphere: A sphere is a solid generated by the revolution of a semi
 
• Curved Surface Area = 
• Total Surface Area = 
• Volume = 
3
4
p
                    Where, r = radius
 
Spherical shell: A spherical shell is a generalization of an annulus to three dimensions
• Volume =
3
4
p
                  Where, 
1
r and 
2
r are its 
• Total Surface Area
 
Hemisphere: A plane passing through the centre
each called a hemisphere.    
• Curved Surface Area = 
• Total Surface Area = 
• Volume = 
3
2
p
                   Where, r = radius.
 
 
 
 
 
 
SURFACE AREAS AND VOLUMES
right circular cone is a solid generated by the revolution of a right 
angled triangle about one of its side containing the right angle as axis.                    
 
Curved Surface Area = rl p 
=
2 2
h r r + p 
Total Surface Area = ) ( r l r + p 
h r
2
p 
, r = radius of base. 
 
l = slant height 
sphere is a solid generated by the revolution of a semi-circle about its diameter.
Curved Surface Area = 
2
4 r p                                                     
Total Surface Area = 
2
4 r p 
3
r p 
, r = radius 
A spherical shell is a generalization of an annulus to three dimensions
) (
3
2
3
1
r r - 
are its external and internal radii respectively.
Total Surface Area = ) ( 4
2
2
2
1
r r + p 
A plane passing through the centre of a sphere cuts the sphere in two equal parts 
 
Curved Surface Area = 
2
2 r p                                                     
Total Surface Area = 
2
3 r p 
3
r p 
, r = radius. 
S 
right circular cone is a solid generated by the revolution of a right 
                     
circle about its diameter. 
                                                      
A spherical shell is a generalization of an annulus to three dimensions. 
respectively. 
cuts the sphere in two equal parts 
                                                      
 
 
 SURFACE AREAS AND VOLUME
 
 
Frustum of a Cone: If a cone is cut by a plane parallel to the base of the cone, then the portion 
between the plane and base is called the 
• Curved Surface Area
                Where, 
1
2
(r h l + =
• Total Surface Area = 
• Volume = 
3
1
h p
                   Where, h    = vertical height
                                 l   = slant height
                             =
2 1
,r r radii of the two bases.
 
 
Surface Area of a Combination of solid:
the curved surface areas of each of the individual parts.
Example: 
 
 
 
 
 
 
 
 
TSA of new solid = CSA of one hemisphere + CSA of cylinder +
Where, TSA= Total Surface Area
and CSA=Curved Surface Area 
 
Volume of a Combination of Solids: 
two basics solids will actually be the sum of the volumes of the constituents.
 
 
                                                                                          
                                   
 
SURFACE AREAS AND VOLUME
If a cone is cut by a plane parallel to the base of the cone, then the portion 
base is called the frustum of a cone. 
Curved Surface Area = ) (
2 1
r r l + p   
2
2 1
) r -                                                    
Total Surface Area = ) ( ) (
2
2
2
1 2 1
r r r r l + + + p p 
) (
2 1
2
2
2
1
r r r r + + 
= vertical height 
= slant height 
radii of the two bases. 
Surface Area of a Combination of solid: The total surface area of the new solid is the sum of 
the curved surface areas of each of the individual parts. 
of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere
Where, TSA= Total Surface Area 
CSA=Curved Surface Area  
Volume of a Combination of Solids: The volume of combination of the solid formed by joining 
will actually be the sum of the volumes of the constituents.
                                                                                           
SURFACE AREAS AND VOLUME 
If a cone is cut by a plane parallel to the base of the cone, then the portion 
The total surface area of the new solid is the sum of 
CSA of other hemisphere 
ombination of the solid formed by joining 
will actually be the sum of the volumes of the constituents. 
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