Class-IX Unit-13 Surface Area and Volume

Download Solution (Surface Area and Volume)

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. m² ,  km²

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	108  squaremeter

Volume

1 cm3	1mL	1000 mm3
1 Litre	1000mL	1000 cm3
1 m3	106  cm3	1000 L
1 dm3	1000 cm3	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 ) H
Diagonal of the cuboids	(L2+B2+H2)1/2
Volume of a cuboids	LBH
Length of all 12 edges of the cuboids	4 (L+B+H).
Surface Area of Cube of side L	6L2
Lateral surface area of the cube	4L2
Diagonal of the cube
Volume of a cube	L3

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	2πrh
Total surface area of cylinder	2πr (h+r)
Volume of Cylinder	π r2h

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	πrL
Total surface area of cone	πr (L+r)
Volume of Cone	(1/3)πr 2h

Surface Area and Volume of sphere and hemisphere 

phere	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	4πr2
Volume of Sphere	(4/3)πr 3
Curved Surface area of hemisphere	2πr2
Total Surface area of hemisphere	3πr2
Volume of hemisphere	(2/3)πr 3
Volume of the spherical shell whose outer and inner radii and ‘R’ and ‘r’ respectively	(2/3)π(R3-r3)

How the Surface area and Volume are determined

Area of Circle	The circumference of a circle is 2πr. 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=(1/2)2πrr=πr2
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=(1/2)2πrs=πrs

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 =2π.r.h==5.5 m²
Total Surface area=2π.r.h + 2πr² =5.89m²
Volume =πr³h=.17 m³
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 cm²
Lateral surface area of cuboidal box  is given by  2[lh + bh]
= [2(12.5 × 8 + 10 × 8)] cm²
= (2 × 180) cm²
= 360 cm²
It is apparent from the data, 400 > 360
The difference =Lateral surface area of cubical box − Lateral surface area of cuboidal box = 400 cm² − 360 cm2 = 40 cm²
So Lateral surface area of cubical box is greater than Lateral surface area of cuboidal box by 40 cm²
 (ii) Total surface area of cubical box = 6(a)² = 6(10 cm)² = 600 cm²
Total surface area of cuboidal box
= 2[lh + bh + lb]
= [2(12.5 × 8 + 10 × 8 + 12.5 × 10] cm²
= 610 cm²
It is apparent from the data, 610 > 600
Difference=Total surface area of cuboidal box − Total surface area of cubical box = 610 cm² − 600 cm² = 10 cm²
So Lateral surface area of cuboidal box is greater than Lateral surface area of cubical box by 10 cm²