AREA OF RECTANGLE

To find area of rectangle we will count the number of squares, total number of squares will be the area of rectangle which can also be obtained by multiply length and breadth.

Area of rectangle=l x b

AREA OF PARALLELOGRAM

To find the area of parallelogram we will cut the ||gm as shown in figure in two parts and when we place 1st and 2nd part as shown in figure it becoms a rectangle and area of rectangle is (l x b )So

Area of ||gm=base x height

AREA OF TRIANGLE

To find out area of triangle we take two triangles of same size and when we place these two triangles as

shown in figure it becomes a ||gm Area of || gm = base x height. Here we have taken two triangles of same size.

Area of one triangle

= ½ area of ||gm.Area of triangle

= ½ base x height

AREA OF TRAPEZIUMTo find out the area of trapezium we take two trapezium of same size and when we place these two trapezium as shown in figure we get a ||gm. If

we name parallel sides as a and b and height h then

Area of ||gm=(a+b)hAREA OF TRAPEZIUM=1/2(a+b)h

RHOMBUSDiagonals of rhombus bisect each other at rt angles. Area of rhombus is sum of area of triangle ABC and area of triangle ACD. If we add area of these two triangles we get area of rhombusArea of triangle ABC=1/2 ACxOBArea of triangle ACD=1/2 ACxOD

Area of rhombus=1/2 ACxOB+1/2 ACxOD =1/2 ACxOB+1/2 ACxOD =1/2 AC(OB+OD)=1/2 ACxBD=1/2 d1 x d2 Where d1, d2 are diagonals

Area of rhombus =1/2 d1 x d2

A B

CD

d1

d2

o

QUADRILATERALTo find area of quadrilateral we meet one diagonal and draw perpendiculars on this diagonal from other two points. Now we get two triangles and area of quadrilateral is sum of area of these two triangles.

Area of quadrilateral =1/2 AC(d1 +d2)

SUM OF ANGLES OF TRIANGLESWe take a triangle and color the angles. If we cut these angles and put on a paper we get a straight line.

So sum of angles of triangles=180o

SUM OF ANGLES OF QUADRILATERALWe take a quadrilateral and color the angles.If we cut these angles as shown in figure and put on a paper

we get a circle.SUM OF ANGLES OF QUADRILATERAL=360O

ab

CUBECube can be made by cutting and folding a paper as shown in figure.Lateral surface area of cube is the area of 4 squares on lateral sides except top and bottom ,if side of cube is a then

Lateral surface area of cube=4a2

Total surface area is the sum of lateral surface area and two faces on top and bottom which are square.So

Total surface area of cube=6a2

Volume is cube is the no of cube of unit length in the cube. We see that in 2 inch cube 8 cubes of one inch comes and in 3 inch cube 27 cube come.

So Volume of cube=a3

Area of circle We draw a circle and half of circle is one color and other half in second color. If we cut it in 16 pieces and place these pieces as shown in figure .It becomes a rectangle, whose length is and height is equal to

radius r of circle.

AREA OF CIRCLE=

Perimeter of SquarePerimeter of Square is sum of four equal sides so

Perimeter of Square=4aSimilarly we can find sum of equilateral pentagon as sum of five equal sides and so on.

Perimeter of equilateral Pentagon =5a

a a

a

a

a a

aa

a

a2- b2 We take a square of side a and

make b2 on one side and make 3 Pieces as shown in figure .

we separate these 3 pieces and put

one piece b2 one side and join remaining two pieces as shown in figure.

The area of these two pieces is equal to

(a + b)(a – b) , so if we subtract

b2 from a2.we get area

(a + b ) (a - b )Hence

a2- b2 =(a+ b) (a-b) .

Area Of 5 piecesArea of 5 pieces is (a + b)2 in middle we get area (a - b)2 if we subtract (a-b)2

From (a + b )2 we get the area 4abHence

(a + b )2 – (a- b )2 =4ab

a

a

a

a

b

b

b

b

(a- b )2

ab

ab

ab

ab

(a+b)2 =a2+b2+2ab To find (a+b +c)2 the value of

(a+b)2 =a2+b2+2ab will remain same and 2ac and 2bc and c2 will be added as shown in figure:-

(a+b +c)2 =a2+b2+2ab +c2+2ac+2bcTo find (a+b +c+d)2 the value of (a+ b +c)2 i.e. (a+ b +c)2 = a2+b2+2ab +c2+2ac+2bc will remain same and d2,2ad,2bd,2cd will be added as shown in fig (3).

(a+b +c+d)2 = a2+b2+2ab +c2+2ac+2bc +d2

+2ad+2bd+2cd

(a-b)2

To find (a-b)2 we take a square board or

paper of side a we take one length equal to

b on two sides of board or paper and make 3 pieces as shown in figure

These 3 Pieces area is a2

We separate these three pieces

One piece is (a-b)2 one is ab and one is b(a-b)If we subtract two pieces ab,b(a-b) from three

pieces we get (a-b)2 and so

(a-b)2 =a2 –ab-b(a-b) =a2 –ab-ab+b2

(a-b)2 =a2 –2ab+b2

CuboidLateral surface area of cuboid is the area of

4 sides except top and bottom

Lateral surface area of cuboid=2lh+2bhTotal surface area of cuboid is lateral surface area and area of top and bottom which is equal to 2lb

Total surface area of cuboid=2lh+2bh+2lbVolume of cuboid is the no. of cubes of unit length that comes in cuboid.

If length of cuboid 5 breadth 3 height 2 we see that 30 cuboid of unit length comes in the cuboid so volume is

30 and product of 5,3,2=30 so

Volume of cuboid=lbh

CylinderCurved surface area of cylinder is area of rectangle whose one side is 2r (circumference of circle)

and other side is h (height of cylinder)

Area of rectangle =2 rxhCurved surface area of cylinder=2 rhTotal surface area of cylinder is sum of curved surface area and area of two circles (2r2)

Total surface area of cylinder=2 rh+ 2r2

Volume of cylinder is product of base r2 and height

h

Volume of cylinder= r2h

PERIMETER OF RECTANGLE

As opposite sides of rectangle are equal if these

sides be l & b then

PERIMETER OF RECTANGLE =2 l +2 b

Perimeter of quadrilateral

Perimeter of quadrilateral=sum of all four sides

Perimeter of quadrilateral=a+b+c+d

PerimeterPerimeter:-Perimeter of any shape is the sum of all outer sides.If we add all outer sides we get the perimeter. If sides are not given we can measure sum of outer sides with the help of thread by placing the thread on one end and revolving the thread upto that end.Perimeter of triangle is sum of all sides as shown in figer .Perimeter of equilateral triangle is sum of three equal sides so

Perimeter of Equilateral Triangle=3aPerimeter of Triangle=a+b+c

Area of square

To find area students will count number of squares of unit length as we see in figure first square is of length 1, second of length 2 ,third of

length 3 and so onArea of square will be number of squares made in each square.

Area of square =a2 or side square

a3+b3

To find the formula of a3 +b3 Let a=3, b=2,take a3 (27 cubes) b3 (8 cubes) place these cubes as shown in figure(1).

Now take upper 22 cubes and put them as shown in figure (2).In rectangle form whose one side

is a+b (3+2=5) other side is a2 +b2 –ab(32 +22-3x2=7)Area of rectangle is

(a+b)(a2 +b2 –ab )33 +23=(3+2)(32 +22-3x2=7) a3 +b3 =(a+b)( a2 +b2 –ab)

a3-b3

To find the formula of a3 -b3 Let a=3, b=2,take a3 (27 cubes) from these 27 cubes take b3 (8 cubes) and place one side the remaining cubes will be a3 -b3 (27-8=19 )These 19 cubes make a rectangle whose one side is a-b (3-2=1) other side is a2 +b2 +ab(32 +22 +3x2=19)Area of rectangle is

(a-b)(a2 +b2 +ab )33 -23=(3-2)(32 +22+3x2=19) a3 -b3 =(a-b)( a2 +b2 +ab)

(a-b)3

Let a=5,b=2 place a3 (53 =125) cubes as shown in figure. Take upper b3 i.e.(23 =8)cubes and put them one side.

Take (a-b)3 i.e.[(5-2)3 =27 ]cubes and put them one side.The remaining cubes are (125-8-27=90)These 90 cubes make a rectanglewhose one side is 3ab(3x5x2=30)And other side is a-b(5-2=3)Whose one side is 3ab(a-b)Now,125=8+27+30x3a3 = b3 +(a- b)3+3ab(a-b)

So, (a-b)3 =a3 - b3 -3ab(a-b)

(a+b)3

Let a=3,b=2 place (a+b)3 i.e (53

=125) cubes as shown in figure.

Take upper a3 i.e.(27cubes) and put them one side.Take b3 (8cubes) and put them one side.The remaining cubes will be (125-8-27=90)These 90 cubes make a rectanglewhose one side is 3ab(3x3x2=18)And other side is (a+b)i.e(3+2=5)Whose area is 3ab(a+b)

Now,53 = 33 + 23+3x3x2(3+2)

So,(a+b)3 =a3 + b3+3ab(a+b)