## Volume and Surface Area of Solid Figures

Surface area related to the amount of space covering the outside of a three-dimensional shape.
Volume and surface area are related to bodies which is solids or hollow.
these bodies occupy space and have generally three dimensions length,breadth and height.

### Surface Area

Surface area is the region covered by the closed plane figure.surface area of a plane figure is the area of all of its surfaces together.It is measured in square unit for example square meter,square centimeter etc.

#### Lateral or curved surface area

Area of solid without its top and bottom is called lateral or curved surface area.

#### Total surface area

Total surface area is the area of whole surface of any object including the area of base and area of top.

### Volume

Volume is the space occupied by an object is called the volume of that particular object.
It is measured in cube unit like cubic centimeter, cubic meter etc.

#### Cube

A three dimensional figure having 6 square faces, 8 vertices and 12 edges with equal length, breadth and height is called cube.

1.Lateral surface area = Perimeter of base × Height
= 4a × a
L.S.A = 4a2

2.Total surface area = Lateral surface area + area of top and area of bottom
= 4a2 + 2a2
T.S.A = 6a2

3.Diagonal = √3a

4.Perimeter = 12a

5. Volume = Area of base × Height
= a2 × a
Volume = a3
Where, a = side of cube

Ex- The diagonal of a cube is 9√3 cm. Find its lateral surface area and volume.
Solution:- Let the side of cube = a
given, Diagonal = 9√3 cm.
lateral surface area and volume = ?
According to question,
Diagonal = 9√3 cm.
or, √3a = 9√3 or, a = 9√3 /√3 = 9
∴ Lateral surface area = 4a2
= 4 ( 9 )2
= 4 × 9 × 9
= 324 sq cm
∴ Volume = a3
= ( 9 )3
= 9 × 9 × 9
= 729 cm 3

#### Cuboid

A three dimensional figure having 6 rectangular faces, 8 vertices and 12 edges with different length breadth and height.
Let Length = l
height = h

1.Lateral surface area = Perimeter of base × height
= 2 ( l + b ) × h
LSA = 2 ( l + b ) h
2. Total surface area = Lateral surface area + area of top and bottom
= 2 ( l + b ) h + 2lb
= 2lh + 2bh + 2lb
= 2 ( lh + bh + lb )
TSA = 2 ( lb + bh + lh )
3. Diagonal = √ ( l 2 + b2 + h2 )
4. Perimeter = 4 ( l + b + h )
5. Volume = Area of base × Height
= ( l × b ) × h
= l × b × h
Volume = lbh

Ex- Find the volume and surface area of a cuboid 17 m long, 13 m broad and 5 m height.
Solution:- given Length = 17 m
breadth = 13 m
height = 6 m
Volume = lbh
= 17 × 13 × 6
= 1326 m 3
Surface area = 2 ( lb + bh + lh )
= 2 ( 17 × 13 + 13 × 6 + 6 × 17 )
= 2 ( 121 + 78 + 102 )
= 2 × 301
= 602 sq m

Ex- A wooden box measures 12 cm × 8 cm × 7 cm. Thickness of wood is 1.5 cm. Find the volume of the wood required to make the box.
Solution:- given, External length = 12 cm
breadth = 8 cm
height = 7 cm
External volume = Length × Breadth × Height
= 12 × 8 × 7
= 672 cm 3
Internal length = 12 - ( 2 × 1.5 ) = 9 cm
breadth = 8 - ( 2 × 1.5 ) = 5 cm
height = 7 - ( 2 × 1.5 ) = 4 cm Internal volume =Length × Breadth × Height
= 9 × 5 × 4
= 180 cm 3
∴ Volume of wood = External volume - Internal volume = 672 - 180
= 492 cm 3

Ex- Find the length of the largest pole that can be placed in a room 12 m long, 8 m broad and 10 m height.
Solution:- given, Length = 12 m
breadth = 8 m
height = 10 m
Length of the longest pole = Length of the diagonal
= √ ( l 2 + b2 + h2 )
= √ ( 122 + 82 + 102 )
= √( 144 + 64 + 100 ) = √308
= 4√77 m

#### Cylinder

A cylinder is a three dimensional shape with circular shapes at top and base and two parallel lines connecting the circular ends.

1.Curved surface area = Perimeter of base × Height
= 2πr × h
CSA = 2πrh 2. Total surface area = Curved surface area + Area of top and base
= 2πrh + 2πr 2
TSA = 2πr ( r + h )
3. Volume = Area of base × Height
= πr 2 × h
V = πr 2h
Where, r = radius of the cylinder
h = height of the cylinder

#### Hollow Cylinder

1.Curved surface area = Curved surface area of outer surface + curved surface area of inner surface
= 2πRh + 2πrh
CSA = 2πh ( R + r )
2. Total surface area = Curved surface area + Area of top and base
= 2πh ( R + r ) + 2π ( R2 - r 2 ) TSA = 2πh ( R + r ) + 2π ( R + r ) ( R - r )
3. Volume = Volume of outer cylinder - Volume of inner cylinder
= πR 2h - πr 2h
= πh ( R 2 - r 2 )
V = πh ( R + r ) ( R - r )

Ex- Find the volume, curved surface area and the total surface area of a cylinder with base radius of 7 cm and height 50 cm.
Solution:- given, radius = 7 cm
height = 50 cm
Volume = πr2h
= ( 22/7 ) × 72 × 50
= ( 22/7 ) × 7 × 7 × 50
= 22 × 7 × 50
= 7700 cm 3
Curved surface area = 2πrh
= 2 × ( 22/7 ) × 7 × 50
= 2 × 22 × 50
= 2200 cm2
Total surface area = 2πr ( r + h )
= 2 × ( 22/7 ) × 7 ( 7 + 50 )
= 2 × 22 × 57
= 2508 cm 2

Ex- A hollow cylinder made of wood has thickness 2 cm while its external radius is 5 cm. if the height of the cylinder is 10 cm,then find the volume and curved surface area of the cylinder.
Solution:- given, R = 5 cm
thickness of wood = 2 cm
height = 10 cm
inner radius = outer radius - thickness
r = 5 - 2
∴ r = 3 cm
Volume of wood = πh ( R 2 - r 2 )
= ( 22/7 ) × 10 ( 52 - 32 )
= ( 220/7 ) × ( 25 - 9 )
= ( 220/7 ) × 16
= 1452/7 cm3

#### Cone

A three dimensional figure having one circular base and one vertex. It is solid or hollow. It is formed by the rotation of a triangle along any of the side.

1.Slant height ( l ) = √( r 2 + h 2  )
2. Curved surface area = πrl = πr√( r 2 + h 2 )
3. Total surface Area = curved surface area + Area of base
= πrl + πr 2
TSA = πr ( l + r )
4. Volume = ( 1/3 ) × Base Area × Height
V = ( 1/3 ) × πr 2 × h
Where , r = radius of base
h = height of the cone
l = slant height

Ex- The curved surface area of a right circular cone of radius 14 cm is 440 sq cm. Find the slant height of the cone.
Solution:- given, radius = 14 cm
curved surface area = 440 sq cm
curved surface area = 440
or, πrl = 440
or, ( 22/7 ) × 14 × l = 440
or, 44 &time;s l = 440
or, l = 440/44
∴ l = 10 cm

Ex- The base radius and height of a cone are 7 cm and 24 cm respectively. Find whole surface area and volume of the cone.
Solution:- given, radius = 7 cm
Height = 24 cm
l = √( r2 + h2 )
= √( 7 2 + 24 2 )
= √( 49 + 576 ) = √ 625
∴ l = 25 cm
Total surface area = πr ( l + r )
= ( 22/7 ) × 7 ( 25 + 7 )
= 22 × 32
= 704 cm2
Volume = ( 1/3 ) × πr2 × h
= ( 1/3 ) &time;s ( 22/7 ) × 72 × 24
= 22 × 7 × 8
= 1232 cm 3

#### Frustum of cone

If the part of solid such as cone or pyramid cut by a plane parallel to the base so as to divide the cone into two parts upper part and lower part, then the lower part is called frustum.

Slant height ( l ) = √{ h2 + ( R - r )2 }
Curved surface area = π ( R + r )l
= π ( R + r )√{ h2 + ( R - r )2 }
Total surface area = π{ rl + Rl + r2 + R2 } Volume = ( rh/3 ) × ( r2 + R2 + rR )
where, r = radius of top
R = radius of base
h = Height
l = slant height

Ex- The a frustum of a right circular cone has a radius of base and top 5 cm and 3 cm respectively and a height of 21 cm. Find the area of its whole surface and volume.
Solution:- given, R = 5 cm
r = 3 cm
h = 21 cm
l = √{ h2 + ( R - r )2 }
= √ { 212 + ( 5 - 3 )2 }
= √( 441 + 4 )
= √445
= 21.01 (approx.)
Total surface area = π{ ( rl + Rl ) + r2 + R2 } = ( 22/7 ) × { ( 3 × 21.01 + 5 × 21.01 ) + 32 + 52 } = ( 22/7 ) × { 63.03 + 105.05 + 9 + 25 }
= ( 22/7 ) × 202
= 22 × 29
= 638 cm2
Volume = ( πh/3 ) × ( r2 + R2 + rR )
= ( 22/7 ) × ( 21/3 ) × ( 32 + 52 + 3 × 5 )
= ( 22/7 ) × 7 × (9 + 25 + 15 )
= ( 22/7 ) × 7 × 49
= ( 1/3 ) × 7 × 22
= 22 × 21 × 49
= 22638 cm 3

Ex- The frustum of a right circular cone has the diameters of base 20 cm, of top 12 cm and height of 10 cm. Find its slant height.
Solution:- given, diameter of base = 20 cm R = 20/2 = 10 cm
diameter of top = 12 cm r = 12/2 = 6 cm
height = 10 cm
Slant height ( l ) = √{ h2 + ( R - r )2 }
= √{ 102 + ( 10 - 6 )2 }
= √{ 100 + 42 }
= √{ 100 + 16 } = √ 4 × 29
= 2√ 29 cm

### Sphere

A three-dimensional solid figure, all points of which are equidistant from a fixed point. That fixed point is center and constant distance is radius of the sphere.

Surface area = 4 πr2
Volume = ( 4/3 ) × πr3
Where, r = radius of the sphere

Ex- Find the volume of sphere of diameter 42 cm.
Solution:- given, diameter = 42 cm
radius = 42/2 = 21 cm
Volume = ( 4/3 ) × πr3
= ( 4/3 ) × ( 22/7 ) × ( 21 )3
= 4 × 22 × 21 × 21
= 88 × 441
= 38808 cm3

Ex- Find the surface area of a sphere of radius 7 cm.
Solution:- given, radius = 7 cm
Area = 4 πr2
= 4 × ( 22/7 ) × ( 7 )2
= 4 × 22 × 7
= 88 × 7
= 616 cm2

#### Hollow Sphere

Internal surface area = 4 πr2
External surface area = 4 πR2
Volume = External volume - Internal volume
= ( 4/3 )>πR3 - ( 4/3 )>πr3
V = ( 4/3 )π ( R3 - r3 )
where, R = External radius
r = Internal radius

Ex- Find the volume of hollow sphere whose outer and inner radius is 4 cm and 2 cm respectively.
Solution:- given, R = 4 cm
r = 2 cm
V = ( 4 π/3 )( R3 - r3 )
= ( 4/3 )× ( 22/7 ) ( 43 - 23 )
= ( 88/21 )× ( 64 - 27 )
= ( 88/21 )× 37
= 234.66 cm3

#### Hemisphere

Hemisphere is the half part of a sphere. when sphere divide into two equal half each half is called hemisphere.

Curved surface area = 2 πr2
Total surface area = 3 πr2
Volume = ( 2πr3 ) / 3
Where, r = radius of the sphere

Ex- Find the curved surface area, total surface area and volume of a hemisphere of radius 42 cm.
Solution:- given, radius = 42 cm
Curved surface area = 2 πr2
= 2 × ( 22/7 ) × 422
= 2 × 22 × 42 × 6
= 11088 cm2
Total surface area = 3 πr2
= 3 × ( 22/7 ) 422
= 3 × 22 × 42 ×6
= 16632 cm2
Total surface area = 3 πr2
= 3 × ( 22/7 ) 422
= 3 × 22 × 42 ×6
= 16632 cm2
Volume = ( 2πr3 ) / 3
= ( 2 × 22 × 423 ) / ( 3 × 7 )
= 2 × 22 × 2 × 42 × 42
= 155232 cm3

### Prism

A three-dimensional figure whose top and base are equal and parallel to each other. whose sides are parallelograms.

1.Lateral surface area = Perimeter of base × Height of prism
2. Total surface area of prism = Lateral surface area + Area of base and top 3. Volume = Area of base × Height

Ex- The base of right prism is a square having side 12 cm. It its height is 9 cm. Find the surface area, total surface area and volume of the prism. Solution:- given, side = 12 cm
height = 9 cm
Lateral surface area = Perimeter of base × Height
= ( 4 × 12 ) × 9
= 432 cm2
Total surface area = Lateral surface area + area of base and top
= 432 + 2 × Area of square
= 432 + 2 × 122
= 432 + 2 × 144
= 432 + 288
= 720 cm2
Volume = Area of base × Height
= ( 12 )2 × 9
= 144 × 9
= 1296 cm2

### Pyramid

A solid whose base is a polygon and top is a point and whose faces are triangles is called a pyramid.

Lateral surface area = ( 1/2 ) × Perimeter of base × Slant height Total surface area = Lateral surface area + Area of the base
Volume = ( 1/3 ) × Area of base × Height

Ex- The base of a pyramid is a square whose side 8 cm, slant height 15cm and vertical height is 18 cm. Find the curved surface area, total surface are and volume of the pyramid.
Solution:- given side of square = 8 cm
slant height = 15 cm
vertical height = 18 cm
Lateral surface area = ( 1/2 ) × × Perimeter of base × Slant height
= ( 1/2 ) × × ( 4 × 8 ) × 15
= 16 × 15
= 240 cm2
Total surface area = Lateral surface area + Area of the base
= 240 + 4 × side2
= 240 + 4 × ( 8 )2
= 240 + 4 × 64
= 240 + 256
= 496 cm2
Volume = ( 1/3 ) × × Area of base × Height
= ( 1/3 ) × ( 8 × 8 ) × 18
= 64 × 6
= 384 cm2

Ex- Find the total surface area of a pyramid having a slant height of 8 cm and a base which is a square of side 4 cm.
Solution:- given, l = 8 cm
side of base = 4 cm
Lateral surface area = ( 1/2 ) × Perimeter of base × Slant height
= ( 1/2 ) × ( 4 × 4 ) × 8
= 16 × 4
= 64 cm2
Total surface area = Lateral surface area + Area of the base
= 64 + 42
= 64 + 16
= 80 cm2