## Saturday, January 2, 2010

### MENSURATION PROJECT

ACKNOWLEDGEMENT
It is a matter of pleasure and privilege to acknowledge my profound gratitude to all those who helped me I completing this project on mensuration . It would never have been possible without the support of my parents. I am also grateful to my school for allowing me to used the library and internet facility at or according to my will and my teachers for their academic guidences and support.
INTRODUCTION
Mensuration is a branch of mathematics which deals with the surface area and volume of solid figures and areas of plane figures . whenever we look , usually we see solids. The figures that can be drawn on our note books or blackboards are called plane figures. Eg. Rectangles , squares, circles etc. If we cut many of these plane figures of the same shape and size from cardboard sheet and stack them up in a vertical pile . by this we obtain some plane figures such as cuboid , cylinder, cube , etc. In Mensuration , we came to know about the surface area and volume of solid figures .
SOLID
The body occuping space are called solids . the solid bodies ocuurs in various shapes such as : a cuboid, a cube, a cylinder, a cone , a sphere , etc.
VOLUME OF SOLID
The space occupied by a solid body is called its volume .
The units of volume are cubic centimeters or cubic meters .
MEASUREMENT OF AREA AND VOLUME
Length
1 centimeter (cm) = 10 milimeters (mm)
1 decimeter(dm) = 10 centimeter
1 meter(m) = 10 dm
= 100 cm
= 1000mm
1 decameter (dam) = 10 meter
= 1000 cm
1 hectometer (hm) = 10 dam = 100 m
1 kilometer (km) = 1000 meter = 100 dam = 10 hm
1 mynameter = 10 kilometer .
Area
1 cm ² = 1 cm x 1 cm = 10 mm x 10 mm = 100mm²
1 dm² = 1 dm x 1 dm = 10 cm x 10 cm = 100 cm²
1m² = 1 m x1m = 10 dm x 10 dm = 100 dm ²
1dam² = 1dam x 1dam = 10 m x 10 m = 100 m ²
1 hm² = 1 hectare = 1 hm x 1 hm = 100 m x 100m = 10,000m ² = 100 dm ²
1km ² = 1 km x 1 km = 10 hm x 10 hm = 100 hm² or 100hectare .
Volume
1cm³ = 1ml = 1cmx 1cm x 1 cm = 10mm x 10mm x 10 mm = 1000mm³
1 litre = 1000 ml = 1000cm³
1m³ = 1m x 1m x 1m = 100cm x 100cm x 100cm = (10³)² cm³ = 1000 litre = 1 kiloletre .
1 dm ³ = 1000m³ .
1m³ = 1000dm³ .
1km = (10³)³ m³.
Faces
Surfaces of a figure and object is known as its faces .
Edges
Any two adjacent faces of a cuboid meet in a line segment , which is an edge of the cuboid .
Vertex
For any two edges that meet at an end point , there is a third edge , that also meets them at that end point . this point of intersection of three degrees of a cuboid is called the vertex of the cuboid.
Base and lateral faces
Any face of a cuboid may be called the base of the cuboid . In that case , the four aces which meet the base are called the lateral face of the cuboid .
Solid cuboid
A solid cuboid or a cuboid or a cuboidal region in the part of space bounded by the size faces of a cuboid.
Solid cube
A solid cube is the part of the space enclosed by the size faces of the cube.
Axis
The line segment joining the centers of 2 bases is called the axis of the cylinder.
CUBOID
Surface area of cuboid
Area of face ABCD = Area of face EFCD = (lxb) cm²
Area of face AEHD = Area of face BEFC = (bxh)cm²
Area of faceABFE = Area of face DHGC = (lxh)cm²
Total surface area of the cuboid
= sum of the area of the cuboid
= 2(l x b) + 2(b x h) + 2(l x h)cm²
=2(l x b) + (b x h) + (l x h) cm²
= 2(lb+bh+lh)cm²
Lateral surface area of the cuboid
= area of face AEHD + area of face BEGC + area of face ABEF + area of face DHGC
=2(b x h) + 2(l x h)
=2(l + b) x h
perimeter of base x height
Diagonal of the cuboid
= root l² + b² + h²
Length of all 12 edges of the cuboid
=4(l+b+h)
Volume of the cuboid
= area of the rectangular sheet x h
= (l x b) x h
= area of base x height
= length x breadth x height

D C
E
H G
A B

F

CUBE
Surface area of a cube
=2(l x l + l x l + l x l)
=2 x 3l²
=6l²
6(edge)²
Lateral surface area of cube
= 2(I x l + l x l)
= 2(l² +l²)
= 4l²
=4(edge)²
Diagonal of a cube
= root 3l
Length of all 12 sides of the cube
= 12 l
Volume of cube
= l x l x l x l
= l³
= (edge)³
CYLINDER
Base
Each of the circular ends on which the cylinder rests is called base.
Lateral surface area
2пrh
Each base surface area
пr²
Total surface area
2пr(h + r)
Each base surface area of hollow cylinder
п(R² - r²)
Curved surface area of hollow cylinder
2п(R + r) (h + R – r) sq. units
Volume of a right circular cylinder
=Area of base x height
= пr² h
Volume of material in hollow cylinder
=Exterior volume – Interior volume
=пr²h – пr²h
=пh (R² - r²) units
CONE
Base
A right circular cone has a plane end , which is in circular shape. This is called the base of the cone.
Slant height
The length of the line segment joining the vertex to any point on the circular edge of the base is called slant height.
Curved surface area of cone
=1/2 x ore length x radius
=1/2 x 2пr x l
=пrl
Total surface area of cone
=curved surface area + area of base
=пrl + пr²
=пr (l +r)
Slant height
=√r² + h²
Volume of cone
= 1/3 (пr²) x h
=1/3 x area of base x height
Note
3(volume of cone of radius (r) and height (h) = volume of cylinder of
SPHERE
The set of all parts in space which are equidistant from a fixed points is called a sphere.
Diameter
A line segment through the centre of a sphere , and with the end points on the sphere is called the diameter of the sphere.
Surface area of sphere
4пr² sq. units
Volume of sphere
4/3пr³ cubic units
HEMISPHERE AND SPHERICAL SHELL
Hemisphere
A plane through the centre of the sphere divides the sphere into two equal parts , each of which is called a hemisphere.
Spherical shell
The difference of two solid concentric spheres is called a spherical shell.
Curved surface area of hemisphere
2пr² sq. units
Total surface area of hemisphere
=2пr² + пr²
=3пr²sq. units
Volume of hemisphere
2/3 пr³
Total surface area of a hemispherical shell
4пr² sq. units
Volume of spherical shell
4/3п (R³ - r³) cubic units
Volume of hemispherical shell
2/3п (R³ - r³) cubic units
Frustum of the cone
Frustum
If a right circular cone is cut off by a plane parallel to its base , then the portion of the cone between the cutting plane and the base of the cone is called the frustum of the cone.
Height
The height or thickness of a frustum is the perpendicular distance between its two circular bases.
Slant height
The slant height of a frustum of a right circular cone is the length of the line segment joining the extremities of two parallel radii , drawn in the same direction of the two circular bases.
Volume of the frustum
= п/3(r1² + r2² + r3²)h
Lateral surface area
= п (r1 + r2)
Total surface area
П {(r1 + r2) l + r1² + r2²}
Slant height of the frustum
√h² + (r1 – r2)
Height of the cone of which frustum is a part
Hr/r1 – r2
Slant height of the cone of which frustum is a part
Lr1/r1 – r2
Volume of the frustum
h/3{A1 + A2 + √A1 x A2} , where A1 and A2 denote the areas of circular bases of the frustum.
BIBLOGRAPHY
I able to make this project ‘MENSURATION’ with the help of same book and internet sites. I took the reference of books ‘Maths NCERT book of class Xth ‘Maths NCERT book of lXth Mathematics class Xth and lXth by R.D. sharma Mthematics class Xth and lXth by R.S aggarwal. I also called this information from internet by using yahoo! and google! Search engines.