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# 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.

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
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