Mensuration

AREA

Fundamental Concepts

I. Points on triangles:

1. Sum of  the angles of a triangle is 180 degrees.

2 .Sum of any two sides of a triangle is greater than the third side.

3. Pythagoras theorem: In a right angle triangle
(Hypotenuse)^2 = (base)^2 + (Height)^2

4.The line joining the midpoint of a side of a triangle to the opposite vertex is called the

5.The point where the three medians of a triangle meet is called CENTROID.

Centroid divides each of the medians in the ratio 2:1.

6.In an isosceles triangle, the altitude from the vertex bi-sects the base

7.The median of a triangle divides it into two triangles of the same area.

8.Area of a triangle formed by joining the midpoints of the sides of a given triangle is one-fourth of the area of the given triangle.

II.Points on Quadrilaterals:

1.    The diagonals of a parallelogram bisects each other .

2.    Each diagonal of a parallelogram divides it into two triangles of the same area

3.    The diagonals of a rectangle are equal and bisect each other.

4.    The diagonals of a square are equal and bisect each other at right angles.

5.    The diagonals of a rhombus are unequal and bisect each other at right angles.

6.    A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

7.   Of all the parallelograms of a given sides , the parallelogram which is a rectangle has the greatest area.

IMPORTANT FORMULAE

I.1.Area of a rectangle=(length*breadth)

Therefore length = (area/breadth) and breadth=(area/length)

2.Perimeter of a rectangle = 2*(length+breadth)

II.Area of a square = (side)^2 =1/2(diagonal)^2

III Area of four walls of a room = 2*(length + breadth)*(height)

IV         1.Area of the triangle=1/2(base*height)

              2.Area of a triangle = (s*(s-a)(s-b)(s-c))^(1/2), 

                 where a,b,c are the sides of a triangle   and s= ½(a+b+c)

              3.Area of the equilateral triangle =((3^1/2)/4)*(side)^2

              4.Radius of incircle of an equilateral triangle  of side a=a/2(3^1/2)

              5.Radius of circumcircle of an equilateral triangle of side a=a/(3^1/2)

              6.Radius of incircle of a triangle of area del and semiperimeter S=del/S

V.1.Area of the parellogram =(base *height)

    2.Area of the rhombus=1/2(product of the diagonals)

    3.Area of the trapezium=1/2(size of parallel sides)*distance between them

VI 1.Area of a circle =pi*r^2,where r is the radius

     2. Circumference of a circle = 2∏R.

     3. Length of an arc = 2∏Rθ/(360) where θ is the central angle

     4. Area of a sector = (1/2) (arc x R) = pi*R^2*θ/360.

VII. 1. Area of a semi-circle = (pi)*R^2.

        2. Circumference of a semi-circle = (pi)*R.

VOLUME AND SURFACE AREA

I. CUBOID

Let length = 1, breadth = b and height = h units. Then, 

1. Volume = (1 x b x h) cubic units.

2. Surface area= 2(lb + bh + lh) sq.units.

3. Diagonal.=Öl² +b² +h² units

II. CUBE

Let each edge of a cube be of length a. Then,

1. Volume = a³ cubic units.

2. Surface area = 6a² sq. units.

3. Diagonal = √3 a units.

 III. CYLINDER

Let radius of base = r and Height (or length) = h. Then,

 1. Volume = (Pi r²h) cubic units.

 2. Curved surface area = (2Pi rh). units.

 3. Total surface area =2Pi.r (h+r) sq. units

 IV. CONE

Let radius of base = r and Height = h. Then,

 1. Slant height, l =√ h²+r²

 2. Volume = (1/3) Pi.r²h  cubic units.

 3. Curved surface area = (Pi.rl) sq. units.

 4. Total surface area = (Pi.rl + Pi.r² ) sq. units.

 V. SPHERE

Let the radius of the sphere be r. Then,

1. Volume = (4/3)Pi.r3 cubic units.

     2. Surface area = (4Pi.r²) sq. units.

 VI. HEMISPHERE

Let the radius of a hemisphere be r. Then,

1. Volume = (2/3)Pi.r³ cubic units.

    2. Curved surface area = (2Pi.r²) sq. units.

    3. Total surface area = (3Pi.r²) units.

        Remember: 1 litre = 1000 cm3.