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Wednesday, 12 July 2017

Mensuration

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MENSURATION

Definition: Mensuration is a science of measurement of the lengths of lines, areas of surfaces and volumes of solids.
                                                                                     
MensurationFlow
                                                                                                                             
Plane Figures: Planes are two-dimensional i.e. these two dimensions are namely length and breadth. These occupy surface.
E.g. Triangle, Quadrilateral & Circle etc.

Triangle: 
                                             
Scalene :
    

a, b and c are three sides of triangle and s is the semi perimeter .
                  S = mensuration , b is the base and h is the altitude of triangle.
                        Area : (i) mensuration
                                   (ii) mensuration ( Hero’s formula)
               Perimeter : a+b+c = 2s


Equilateral :
                       a = B Side,
                       h = Height or altitude,
                       h = mensurationa
                      Area : (i) mensuration (ii) mensurationa2
                     Perimeter : 3a


Isosceles :
                             a = Equal sides
                             b =  Base
                            h = Height of altitude
                          h = mensuration
                         Area : (i) mensuration
                                 (ii) mensuration
                    Perimeter: 2a + b


Rectangle: 

          l = Length
              b = Breadth
              d = Diagonal
               Area :       mensuration
      Perimeter :  2l+2b = 2(l +b)
       Diagonal :  mensuration


Square :
a = Side
                         d = Diagonal
                Diagonal :  mensuration
                  Area : (i)    a × a = a2
                                           (ii)    mensuration
         Perimeter : a + a + a + a = 4a


Parallelogram :
            

a and b are sides adjacent to each other.
              h= Distance between the parallel sides
                             Area :        a × h
                         Perimeter :      2 (a + b)


Rhombus : 


            a = Each equal side of rhombus
            d1 and d2 are the diagonals
                                    d1 = BD
                                    d2 =  AC
                                    Area :     mensuration d1× d2
                         Perimeter :     4a


Trapezium : 


a and b are parallel sides to each other and h is the perpendicular distance between parallel sides.
                              
                               Area :  mensuration×h
                      Perimeter : AB + BC + CD + AD


Circle :
                                       r =  Radius of the circle
                      Area =  mensuration r2
         Perimeter : 2 mensuration r (Called as circumference)
                              mensuration  =  mensuration = 3.1416   
                                                                      (Approx.)



Ex:   Calculate the area of a triangle whose sides are 8 cm, 6 cm and 10 cm?
 


Sol:   Area of triangle = mensuration
            
                 A = mensuration
                    = mensuration
                  = 24 cm2 


Ex:   The sides of a triangular field are 40 m, 32 m and 24 m respectively. Find the cost of ploughing this field at the rate of Rs. 5 per square metre? 


Sol: A = mensuration
    = mensuration
    = 384 m2
     Total cost = Rate ×Area
                       = 5 × 384
                       = Rs. 1920


Ex:   Find the diagonal of rectangle whose length is 4m and area 12m2 ?


 Sol:          b = mensuration = mensuration = 3m
 Diagonal = mensuration [Using Pythagoras theorem]
                       
                 = mensuration
                 = mensuration
                 = 5m


Ex:   The drawing room of a house is 10 m by 6 m and it is to be covered with a carpet of width 1m 50 cm. Find the length of the carpet required? 

Sol: Area to be covered by carpet = Floor area of the drawing room
                                                = 10 × 6
                                                = 60 m2     
      Length of the carpet required = mensuration
                                                     = mensuration
                                                     = 40 m


Ex: One of the diagonals of a rhombus of side 5 cm measures 8 cm. Find the area of the rhombus? 


Sol: S = 5 cm, d1 = 8 cm
mensuration
=   mensuration
mensuration
mensuration 


EX: The altitude of an equilateral triangle is mensuration cm. What is its perimeter?


Sol:   Let one side of equilateral triangle = a
                    mensuration  h =  mensuration  = mensuration
                    Þ  mensuration =  mensuration  Þ a = 2
              mensuration    Perimeter = 2 x3 = 6 cm.


Ex: The sides of a right angled triangle are equal to three consecutive numbers expressed in centimeters. What can be the area of such a triangle?

Sol: The side of right angled triangle which is 3 consecutive numbers are 3 cm, 4 cm and 5 cm
         ⇒ Then, the area of  triangle = mensuration x Base x Height
                                                      = mensuration x 3 x 4
                                                      = 6 cm2


Ex: ABC is an isosceles triangle such that AB = BC = 8 cm and ÐABC = 900.What is the length of the perpendicular drawn from B on AC?


Sol:
  ABC is an isosceles triangle such that AB = BC = 8 cm and ÐABC = 900
 We know that –
   Area of Triangle ABC –
 
                  mensuration × AB × BC = mensuration × AC × BD
                       8 × 8 = 8mensuration × BD
                             BD = 4mensuration cm.


Ex:  A triangle DEF is formed by joining the midpoints of the sides of triangle ABC similarly a triangle PQR is formed by joining the midpoints of the sides of the triangle DEF. If the sides of the triangle PQR are of lengths 1,2 and 3 units, what is the perimeter of the triangle ABC?  


Sol:   Perimeter of mensuration PQR = 1 + 2+ 3
                                                   = 6 units
          Sides of mensuration DEF are 2, 4 and 6 units , then
          Perimeter of mensuration DEF = 2 + 4 + 6
                                       = 12 units
  Now, Sides of mensuration ABC = 4, 8 and 12 units , then
         Perimeter of mensuration ABC = 4 + 8 + 12
                                       = 24 units


EX: The area of a rectangle whose one side is ‘a’ is ‘2a2’. What is the area of a square having one of the diagonals of the rectangle as side?


Sol:
  Other side of the rectangle = mensuration
                                                  
                                                   =   mensuration = 2a
     Diagonal of the rectangle = mensuration
                                                    = mensuration a
   Area of the square on the diagonal = 5a2

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