## 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 = 4a**^{2}** **

**2**.Total surface area = Lateral surface area + area of top and area of bottom

= 4a^{2} + 2a^{2} ** T.S.A = 6a**^{2}** **

**3**.Diagonal = **√3a**

**4**.Perimeter = **12a **

**5**. Volume = Area of base × Height

= a^{2} × a

Volume = **a**^{3}

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 = 4a^{2}

= 4 ( 9 )^{2}

= 4 × 9 × 9

= 324 sq cm

∴ Volume = a^{3}

= ( 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

breadth = b

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}** + b**^{2}** + h**^{2}** ) **

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} + b^{2} + h^{2} )

= √ ( 12^{2} + 8^{2} + 10^{2} )

= √( 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 **^{2}**h **

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 ( R^{2} - r ^{2} ) **TSA = 2πh ( R + r ) + 2π ( R + r ) ( R - r ) **

3. Volume = Volume of outer cylinder - Volume of inner cylinder

= πR ^{2}h - πr ^{2}h

= 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 = πr^{2}h

= ( 22/7 ) × 7^{2} × 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 cm^{2}

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 ( 5^{2} - 3^{2} )

= ( 220/7 ) × ( 25 - 9 )

= ( 220/7 ) × 16

= 1452/7 cm^{3}

#### 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 = √( r^{2} + h^{2} )

= √( 7 ^{2} + 24 ^{2} )

= √( 49 + 576 ) = √ 625

∴ l = 25 cm

Total surface area = πr ( l + r )

= ( 22/7 ) × 7 ( 25 + 7 )

= 22 × 32

= 704 cm^{2}

Volume = ( 1/3 ) × πr^{2} × h

= ( 1/3 ) &time;s ( 22/7 ) × 7^{2} × 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 ) =** √{ h**^{2}** + ( R - r )**^{2}** } **

**Curved surface area = π ( R + r )l **

= π ( R + r )√{ h^{2} + ( R - r )^{2} }

**Total surface area = π{ rl + Rl + r**^{2}** + R**^{2}** } ** **Volume = ( rh/3 ) × ( r**^{2}** + R**^{2}** + 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 = √{ h^{2} + ( R - r )^{2} }

= √ { 21^{2} + ( 5 - 3 )^{2} }

= √( 441 + 4 )

= √445

= 21.01 (approx.)

Total surface area = π{ ( rl + Rl ) + r^{2} + R^{2} } = ( 22/7 ) × { ( 3 × 21.01 + 5 × 21.01 ) + 3^{2} + 5^{2} } = ( 22/7 ) × { 63.03 + 105.05 + 9 + 25 }

= ( 22/7 ) × 202

= 22 × 29

= 638 cm^{2}

Volume = ( πh/3 ) × ( r^{2} + R^{2} + rR )

= ( 22/7 ) × ( 21/3 ) × ( 3^{2} + 5^{2} + 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 ) = √{ h^{2} + ( R - r )^{2} }

= √{ 10^{2} + ( 10 - 6 )^{2} }

= √{ 100 + 4^{2} }

= √{ 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 πr**^{2}** ** **Volume = ( 4/3 ) × πr**^{3}** **

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 ) × πr^{3}

= ( 4/3 ) × ( 22/7 ) × ( 21 )^{3}

= 4 × 22 × 21 × 21

= 88 × 441

= 38808 cm^{3}

**Ex**- Find the surface area of a sphere of radius 7 cm. **Solution**:- given, radius = 7 cm

Area = 4 πr^{2}

= 4 × ( 22/7 ) × ( 7 )^{2}

= 4 × 22 × 7

= 88 × 7

= 616 cm^{2}

#### Hollow Sphere

**Internal surface area = 4 πr**^{2}** ** **External surface area = 4 πR**^{2}** **

Volume = External volume - Internal volume

= ( 4/3 )>πR^{3} - ( 4/3 )>πr^{3}** V = ( 4/3 )π ( R**^{3}** - r**^{3}** )**

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 )( R^{3} - r^{3} )

= ( 4/3 )× ( 22/7 ) ( 4^{3} - 2^{3} )

= ( 88/21 )× ( 64 - 27 )

= ( 88/21 )× 37

= 234.66 cm^{3}

#### 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 πr**^{2}** **

**Total surface area = 3 πr**^{2}** ** ** Volume = ( 2πr**^{3}** ) / 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 πr^{2}

= 2 × ( 22/7 ) × 42^{2}

= 2 × 22 × 42 × 6

= 11088 cm^{2}

Total surface area = 3 πr^{2}

= 3 × ( 22/7 ) 42^{2}

= 3 × 22 × 42 ×6

= 16632 cm^{2}

Total surface area = 3 πr^{2}

= 3 × ( 22/7 ) 42^{2}

= 3 × 22 × 42 ×6

= 16632 cm^{2}

Volume = ( 2πr^{3} ) / 3

= ( 2 × 22 × 42^{3} ) / ( 3 × 7 )

= 2 × 22 × 2 × 42 × 42

= 155232 cm^{3}

### 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 cm^{2}

Total surface area = Lateral surface area + area of base and top

= 432 + 2 × Area of square

= 432 + 2 × 12^{2}

= 432 + 2 × 144

= 432 + 288

= 720 cm^{2}

Volume = Area of base × Height

= ( 12 )^{2} × 9

= 144 × 9

= 1296 cm^{2}

### 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 cm^{2}

Total surface area = Lateral surface area + Area of the base

= 240 + 4 × side^{2}

= 240 + 4 × ( 8 )^{2}

= 240 + 4 × 64

= 240 + 256

= 496 cm^{2}

Volume = ( 1/3 ) × × Area of base × Height

= ( 1/3 ) × ( 8 × 8 ) × 18

= 64 × 6

= 384 cm^{2}

**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 cm^{2}

Total surface area = Lateral surface area + Area of the base

= 64 + 4^{2}

= 64 + 16

= 80 cm^{2}