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 =
aArea : (i)
(ii) 
a2Perimeter : 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× d2Perimeter : 4a
Trapezium :
a and b are parallel sides to each other and h is the perpendicular distance between parallel sides.
Area :
×hPerimeter : AB + BC + CD + AD
Circle :
r = Radius of the circle
Area =
r2Perimeter : 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 =
= 
= 3mDiagonal =
[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
× BDBD = 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 , thenPerimeter of
DEF = 2 + 4 + 6= 12 units
Now, Sides of
ABC = 4, 8 and 12 units , thenPerimeter 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 =
=
= 2aDiagonal of the rectangle =
=
aArea of the square on the diagonal = 5a2
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MAHENDRA GURU