SURFACE AREAS AND VOLUMES
We can measure surface area and volume of different solid
figures like cube, cylinder, cone, sphere etc.
Surface Area
Surface Area is the area of surface of a figure. In other words,
surface area is the measure of surface area as a plane. Surface
Volume Of Cylinder
The magnitude of space occupied by the cylinder is called
volume of the cylinder.
Volume Of cylinder = Area Of Base × Height
2
= π r2 × h
area is measured as unit .
Volume
Volume is the magnitude of space of a solid figure. In other
words, volume is the space occupied by a solid figure. Volume
3
is measured as unit .
= π r 2h
Two circles with same centre are called concentric circles.
Similarly, two cylinders with same axis are called coaxial
cylinders.
Now, we will study various solid figures and their surface
areas, volumes etc.
Hollow Cylinder
1. Right Circular Cylinder
1.
2.
3.
A right circular cylinder is a solid generated by the revolution
of a rectangle about one of its side.
r
Hollow cylinder is a figure formed by two cylinders such that: The cylinders are coaxial, i.e. they have common axis.
The cylinders have same heights.
The radius of one cylinder is less than radius of the other.
O
R
O
r
h
h
r'
O'
Base Of Cylinder
Two circles with centre O of radius Or and with centre O’ of
radius O’r’ are called bases or ends of cylinder.
R'
r'
O'
Lateral Surface Area = Lateral Surface Area Of Circle1
+ Lateral Surface Area Of Circle2
= 2π Rh + 2 π rh
= 2 πh(R +r )
Area Of Each Base = π r 2
Height Of Cylinder
Perpendicular distance OO’ between two bases of cylinder is
called height or length of cylinder.
Total Surface Area = 2 × Area Of Base Ring +
Lateral Surface Area
[
Lateral Surface Of Cylinder
Curved surface area joining the perimeters of two bases is
called lateral surface or curved surface of cylinder.
Lateral/Curved Surface Area = Perimeter Of Base × Height
= 2 πr ×h
= 2 πrh
Total Surface Area Of Cylinder
Sum of areas of two bases and lateral surface area is called
total surface area of cylinder.
Total Surface Area = (2 × Area Of Base) + Lateral Surface Area
2
= 2 π r + 2 π rh
= 2 π r (r + h)
]
= 2 × (π R 2 − π r 2 ) + 2 π h (R + r )
2
2
= 2 π (R − r ) + 2 π h (R + r )
= 2 (R + r ) (R − r ) + 2π h (R + r )
= 2 π (R + r ) (R − r + h)
Volume = Volume Of Circle1 – Volume Of Circle2
= π R 2 h − π r2 h
= π h (R 2 − r 2 )
Cylinderical Shell
Cylinderical shell is the figure formed by two cylinders such
that: 1.
The cylinders are coaxial, i.e. they have common axis.
2.
3.
The height of one cylinder is less than other cylinder.
The radius of one cylinder is less than radius of the other.
O
R
o
r
h
r'
R'
Lateral Surface Area = π rl
Total Surface Area Of Cone
Sum of area of base and lateral surface area is called total
surface area of cone.
Total Surface Area = Area of base + Lateral
H
= π r 2 + π rl
= πr (r +l)
o'
O'
Outer Lateral Surface = Lateral Surface Area of Outer cylinder
= 2 π RH
Total Outer Lateral Surface Area =
(2 × Area Of Base Of Outer Cylinder)
+ Lateral Surface Area Of Outer cylinder
)
(
= 2 × π R2 + 2 π R H
Volume Of Cone
The magnitude of space occupied by the cone is called volume
of the cone.
1
Volume = × Area of base × Height
3
1
= × π r2 × h
3
1 2
= πr h
3
3. Sphere
2
= 2 π R + 2 π R2 H
= 2 πR (R +H)
A sphere is a solid generated by revolution of a circle about its
centre in all directions.
r4
Volume = Volume Of Outer Cylinder– Volume Of Inner Cylinder
= π R 2 H − π r2 h
= π (R 2H − r 2h)
r1
2. Right Circular Cone
O
r3
A right circulars cone is a solid generated by revolution of a
right triangle about one of its side.
r2
V
Surface Area Of sphere
Whole covered area is called surface area of sphere.
l
r
h
O
Base Of Cone
The circle with centre O and radius or is called base of the
cone.
Surface Area = 4 π r 2
Volume Of Sphere
The magnitude of space occupied by the sphere is called
volume of sphere.
Volume =
4 3
πr
3
Area Of Base = π r 2
Two circles with same centre are called concentric circles.
Similarly, two spheres with same centers are called concentric
spheres.
Height Of Cone
Perpendicular distance OV from centre O of the base to vertex
V is called height of the cone.
Hemisphere
Slant/Lateral Height Of Cone
Distance rV from perimeter of base to vertex V is called slant
height of cone.
Slant Height =
=
Height 2 + Radius 2
A plane passing through centre of the sphere divides the
sphere into two equal parts. Each of these parts is called a
hemisphere.
Area Of Base = π r 2
r1
O
h2 + r 2
Lateral Surface Of Cone
Curved surface area joining the perimeter of base to the vertex
V is called lateral surface area of cone.
r2
r3
1
× Volume Of Sphere
2
1
4
=
×
π r3
2
3
2
= π r3
3
Volume =
O
R
1
× Surface Area Of Sphere
2
1
= × 4π r2
2
r
Curved Surface Area =
Outer Surface Area = Surface Area Of Outer Sphere
= 4 π R2
= 2 π r2
Total Surface Area = Area Of Base + Area Of Curved Surface
Volume Of = Volume Of Outer Sphere – Volume Inner Sphere
4
4
π R 3 − π r3
3
3
4
3
=
π R − r3
3
= π r2 + 2 π r2
=
= 3 π r2
(
Spherical Shell
Spherical shell is a figure formed by two spheres such that: 1. The spheres are cocentric, i.e. they have common centre.
2. The radius of one sphere is less than the other.
)
Lateral /Curved
Surface Area
Total Surface
Area
Volume
Nomenclature
4 a2
6a2
a2
a : Side Of Cube
Cuboid
2h(l+b)
2 ( lb + bh + hl )
lbh
l : Length
b : Breadth
h : Height
Right Prism
Perimeter Of Base
×
Height
Lateral Surface
Area
+
2 (Area Of One
Base)
Area Of Base
×
Height
2 πr h
2πr (r +h)
π r2 h
1
(Perimeter Of Base)
2
×
Slant Height
Lateral Surface
Area
+
Area Of base
1
×
3
Area Of Base
×
Height
πrl
πr (l+r )
1 2
πr h
3
r : Radius Of Base
h : Height
l : Slant Height
4π r 2
4 π r2
4 3
πr
3
r : Radius
Name Of Solid
Figure
a
Cube
a
a
r
Right Circular
Cylinder
h
Right Pyramid
Right Circular
Cone
h
l
r : Radius of Base
h : Height
r
Sphere
r
r
r
Hemisphere
r
2 π r2
3 π r2
2 3
πr
3
r : Radius