__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?

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**

**Ð**

**ABC = 900.What is the length of the perpendicular drawn from B on AC?**

**ABC is an isosceles triangle such that AB = BC = 8 cm and ÐABC = 900**

Sol:

Sol:

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?**

**Other side of the rectangle =**

Sol:

Sol:

= = 2a

Diagonal of the rectangle =

= a

Area of the square on the diagonal = 5a2

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MAHENDRA GURU