Table of Content
Plane Figure
Plane figure are the figure which lies in a plane or to put it simply which we can draw on a piece of paper
Example : Triangle ,circle,quadilateral etc
We have already studied about perimeter and area of the plane figure
Just a recall of them
Solid Figure
Solid figure does not lie in a single plane.They are three dimensional figure
Example:Cube ,Cylinder, Sphere
Mensuration
 It is branch of mathematics which is concerned about the measurement of length ,area and Volume of plane and Solid figure
Perimeter
 The perimeter of plane figure is defined as the length of the boundary
 It units is same as that of length i.e. m ,cm,km
1 Meter

10 Decimeter

100 centimeter

1 Decimeter

10 centimeter

100 millimeter

1 Km

10 Hectometer

100 Decameter

1 Decameter

10 meter

1000 centimeter

Surface Area or Area
 The area of the plane figure is the surface enclosed by its boundary
 It unit is square of length unit. i.e. m^{2} , km^{2}
1 square Meter

100 square Decimeter

10000 square centimeter

1 square Decimeter

100 square centimeter

10000 square millimeter

1 Hectare

100 squareDecameter

10000 square meter

1 square myraimeter

100 square kilometer

10^{8} squaremeter

Volume
1 cm^{3} 
1mL

1000 mm^{3}

1 Litre

1000mL

1000 cm^{3}

1 m^{3}

10^{6 }cm^{3}

1000 L

1 dm^{3}

1000 cm^{3}

1 L

Surface Area and Volume of Cube and Cuboid
Type

Measurement

Surface Area of Cuboid of Length L, Breadth B and Height H

$2( LB + BH + LH )$.

Lateral surface area of the cuboids

$2( L + B ) \times H$

Diagonal of the cuboids

$\sqrt {L^2+B^2+H^2}$

Volume of a cuboids

$L \times B \times H$

Length of all 12 edges of the cuboids

$4 \times (L+B+H)$.

Surface Area of Cube of side L

$6L^2$

Lateral surface area of the cube

$4L^2$

Diagonal of the cube

$L \sqrt {3}$

Volume of a cube

$L^3$

Surface Area and Volume of Right circular cylinder
Radius

The radius (r) of the circular base is called the radius of the cylinder

Height

The length of the axis of the cylinder is called the height (h) of the cylinder

Lateral Surface

The curved surface joining the two base of a right circular cylinder is called Lateral Surface.

Type

Measurement

Curved or lateral Surface Area of cylinder

$2 \pi rh$

Total surface area of cylinder

$2 \pi r (h+r)$

Volume of Cylinder

$ \pi r^2 h$ 
Surface Area and Volume of Right circular cone
Radius

The radius (r) of the circular base is called the radius of the cone

Height

The length of the line segment joining the vertex to the centre of base is called the height (h) of the cone.

Slant Height

The length of the segment joining the vertex to any point on the circular edge of the base is called the slant height (L) of the cone.

Lateral surface Area

The curved surface joining the base and uppermost point of a right circular cone is called Lateral Surface

Type

Measurement

Curved or lateral Surface Area of cone

$ \pi rL$

Total surface area of cone

$ \pi r (L+r)$

Volume of Cone

$ \frac {1}{3} \pi r^2 h$

Surface Area and Volume of sphere and hemisphere
Sphere

A sphere can also be considered as a solid obtained on rotating a circle About its diameter 
Hemisphere

A plane through the centre of the sphere divides the sphere into two equal parts, each of which is called a hemisphere

radius

The radius of the circle by which it is formed

Spherical Shell

The difference of two solid concentric spheres is called a spherical shell

Lateral Surface Area for Sphere

Total surface area of the sphere

Lateral Surface area of Hemisphere

It is the curved surface area leaving the circular base

Type

Measurement

Surface area of Sphere

$4 \pi r^2$

Volume of Sphere

$ \frac {4}{3} \pi r^3$

Curved Surface area of hemisphere

$ 2 \pi r^2$

Total Surface area of hemisphere

$3 \pi r^2$

Volume of hemisphere

$ \frac {2}{3} \pi r^3$

Volume of the spherical shell whose outer and inner radii and ‘R’ and ‘r’ respectively

$ \frac {2}{3} \pi (R^3r^3)$

How the Surface area and Volume are determined
Area of Circle

The circumference of a circle is $2 \pi r$. This is the definition of π (pi). Divide the circle into many triangular segments. The area of the triangles is 1/2 times the sum of their bases, 2πr (the circumference), times their height, r.
$A= \frac {1}{2} \times 2 \pi r \times r= \pi r^2$

Surface Area of cylinder

This can be imagined as unwrapping the surface into a rectangle.

Surface area of cone

This can be achieved by divide the surface of the cone into its triangles, or the surface of the cone into many thin triangles. The area of the triangles is 1/2 times the sum of their bases, p, times their height,
$A= \frac {1}{2} \times 2 \pi r \times s=\pi rs$

How to solve Surface Area and Volume Problem
(1) We have told explained the surface area and volume of various common sold shapes.
(2) Try to divide the given solid shape into known shapes if the solid figure is other than known shapes
(3) Find out the given quantities like radius,height
(4) Apply the formula from the above given tables and get the answer
(5) Make sure you use common units across the problem
Solved example
Question 1
A cylinder is 50 cm in diameter and 3.5 m in height
(a) Find the radius
(b) Find the Curved Surface area
(c) Total Surface area
(d) Volume
Solution
Height of the cylindrical pillar =h=3.5 m
Diameter of the cylindrical pillar =d=50 cm
So Radius of the cylindrical pillar =r=50/2=25 cm =0.25 m
So Curved surface of the cylindrical pillar =$2 \pi \times r \times h=5.5$ m^{2}
Total Surface area= $2 \pi r h + 2 \pi r^2 =5.89$ m^{2}
Volume =$\pi r^2 h=.17$ m^{3}
Question 2
A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.
(i) Which box has the greater lateral surface area and by how much?
(ii) Which box has the smaller total surface area and by how much?
Solution:
(i)
Dimension of Cube
Edge of cube (a) = 10 cm
Dimension of Cuboidal
Length (l) of box = 12.5 cm
Breadth (b) of box = 10 cm
Height (h) of box = 8 cm
Lateral surface area of cubical box is given by 4(a)^{2}
= 4(10 cm)^{2}
= 400 cm^{2}
Lateral surface area of cuboidal box is given by 2[lh + bh]
= [2(12.5 × 8 + 10 × 8)] cm^{2}
= (2 × 180) cm^{2}
= 360 cm^{2}
It is apparent from the data, 400 > 360
The difference =Lateral surface area of cubical box − Lateral surface area of cuboidal box = 400 cm^{2} − 360 cm2 = 40 cm^{2}
So Lateral surface area of cubical box is greater than Lateral surface area of cuboidal box by 40 cm^{2}
(ii) Total surface area of cubical box = 6(a)^{2} = 6(10 cm)^{2} = 600 cm^{2}
Total surface area of cuboidal box
= 2[lh + bh + lb]
= [2(12.5 × 8 + 10 × 8 + 12.5 × 10] cm^{2}
= 610 cm^{2}
It is apparent from the data, 610 > 600
Difference=Total surface area of cuboidal box − Total surface area of cubical box = 610 cm^{2} − 600 cm^{2} = 10 cm^{2}
So Lateral surface area of cuboidal box is greater than Lateral surface area of cubical box by 10 cm^{2}
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