Tuesday, 13 December 2011

IX Maths Surface Areas and Volumes Concepts Chapter 13

A cuboid is a solid bounded by six faces that arerectangular in shape.
A cuboid whose length, breadth and height are equal is called a cube.
Any two faces other than the opposite faces are called adjacent faces.
Any two adjacent faces meet in a line segment, which is called an edge of the cuboid.
The point of concurrency of any three edges of a cuboid is called a vertex of thecuboid.
A cuboid has 8 vertices and 12 edges. Any faceof a cuboid may be called the base of the cuboid.
In that case, the four adjacent faces of the base are called thelateral faces of the cuboid.
The lateral surface area of the cuboid = 2 (l + b) h.
It is equal to sum of areas of all its lateral faces.
Lateral surface area of the cube=4a2
Total surface area of the cuboid=2(lb+bh+lh)
Total surface area of the cube=6a2
Volume of a solid object is the measure of the space occupied by it
The volume of substance that can be stored by the object is called its capacity.
Volume of Cuboid = length x breadth x height=lbh
Volume of a Cube =(edge)3=(a)3
L.S.A of a cuboid =2 (l + b) h
T.S.A of a cuboid =2(lb+bh+lh)
Volume of the cuboid =lbh
T.S.A of a cube =6a2
Total surface area of a cube=sum areas of all the faces of a cube
volume of the cube=a3
A cylinder can be defined as a solid figure that is bound by a curved surface and two flat surfaces. 
The flat surfaces are made up of two congruent circles that are parallel to each other. 
These flat surfaces are called thebases of the cylinder. 
The radius of the circular bases is the radius of the cylinder. 
The perpendicular line that passes through the centers of the two circular bases is the height of the cylinder or axis of the cylinder. 
A cylinder is said to be right circular cylinder if axis is perpendicular to the radius of the cylinder.
The curved surface joining the two basses of a right circular cylinder is called its lateral surface area.
For a right circular cylinder of radius r and height h
Lateral surface area of the Cylinder =  2prh
Base surface area of the Cylinder =pr2 
Total surface area of the Cylinder =2prh+pr2 +pr2=pr(r+h)
Volume of the Cylinder = pr2h
Cone is solid figure with a circular base that tapers to a point or vertex.
 A Cone is said to be a right circular cone if its height is perpendicular to the radius of the base.
Let "r" be the base radius, "h" be the height, "l" be the slant height of a right circular cone.
Then l = √(r2 + h2 )
Base area of the cone = p r2 
Curved surface area of the cone =  p r l
Total surface area of the cone= p r l  + = p r2   =pr (l+r)
Volume of Cone = 1/3 p r2 h
A sphere is a three dimensional figure, made up of points that are equidistant from a given point.
It does not have an edge or a vertex.
The surface of a sphere is uniform and smooth.
The centre of a sphere is a point, which is equidistant from all the points on a sphere.
The distance between the centre and any point on the surface of the sphere is called the radius of a sphere.
Generally the radius is denoted by the letter r.
A line segment through the centre of the sphere, and with the end points on the sphere is called a diameter of the sphere.

IX Maths Surface Areas and Volumes Important Concepts
1. Every cube is a cuboid but every cuboid is not a cube.
2. In a cube, the lengths of all edges are the same. 
3. Area of the four walls of the hall is the lateral surface area of cube or cuboid4. The outer surface of a cuboid is made up of six rectangles or six rectangular regions, called the faces of the cuboid
5. In case of a room, lateral surface area means the area of the four walls of the room, whereas total surface area means the area of
four walls plus the area of the floor and the ceiling.
6. Face diagonal of a cube or cuboid will always be shorter than the body diagonal.
7. The total surface area of any object will be greater than its lateral surface area.
8. The unit of measurement of both volume and capacity is cubic unit such as cubic feet, cubic cm. cubic m etc.
9. When an object of certain volume is recast into a cylinder, the volume of the cylinder formed will always be equal to the volume of the original object.

10. The solids having the same curved surface do not necessarily occupy the same volume.

11. When an object is dropped into a liquid, the volume of the displaced liquid is equal to the volume of the object that is dipped.
12. Volume of a Cone is one third of the volume of a cylinder of same height and radius of base

13. Volume of a hemisphere be exactly half of the corresponding sphere.
14. All the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume

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