Impotant Notes : Mensuration

Definition: Mensuration is a science of measurement of the lengths of lines, areas of surfaces and volumes of solids.
                                                                                     

                                                                                                                             
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 =  , b is the base and h is the altitude of triangle.
                        Area : (i)  
                                   (ii)  ( Hero’s formula)
               Perimeter : a+b+c = 2s


Equilateral : 
                       a = B Side, 
                       h = Height or altitude, 
                       h = a
                      Area : (i)  (ii) a2
                     Perimeter : 3a

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

Rectangle: 

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

Square : 
a = Side
                         d = Diagonal
                Diagonal :   
                  Area : (i)    a × a = a2
                                           (ii)     
         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 :      d1× d2
                         Perimeter :     4a

Trapezium : 

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

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


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

Sol:   Area of triangle = 
             
                 A =  
                    =  
                  = 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 =  
    = 
    = 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 =  =  = 3m
 Diagonal =  [Using Pythagoras theorem]
                        
                 =  
                 = 
                 = 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 =  
                                                     =  
                                                     = 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
 
=    
 
 

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


Sol:   Let one side of equilateral triangle = a
                      h =    =  
                    Þ   =    Þ a = 2 
                  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 =  x Base x Height
                                                      =  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 –
  
                   × AB × BC =  × AC × BD 
                       8 × 8 = 8 × BD
                             BD = 4 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  PQR = 1 + 2+ 3 
                                                   = 6 units
          Sides of  DEF are 2, 4 and 6 units , then
          Perimeter of  DEF = 2 + 4 + 6
                                       = 12 units
  Now, Sides of  ABC = 4, 8 and 12 units , then
         Perimeter of  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 =  
                                                   
                                                   =    = 2a
     Diagonal of the rectangle = 
                                                    =  a
   Area of the square on the diagonal = 5a2