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

=2(length x breadth+breadth x height+height x length)

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.

Radius

The radius of circular base is called radius of cylinder.

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

Radius (r) & height (h)

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.

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