Q1 ) A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a. Find the area of the signal board, using heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

We know that an equilateral triangle has equal sides.

So, all sides are equal to a.

If Perimeter of triangle = 180 cm (Given)

(Since, 2s = a + b + c)

We also, know that, Area of an equilateral triangle

=

(Since, Heron's formula =

=

=

=

Q2 )
The triangular side walls of a flyover have been used for advertisements. The sides of
the wails are 122 m, 22 m and 120 m (see figure). The advertisements yield earnings of Rs. 5000 per

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Let a = 122 m, b = 22 m, c = 120 m

Also, we have,

Thus, we observe that the side walls are in right triangular shape.

Thus, the area of the triangular side walls

=

We know that, yearly rent = Rs.5000 per

Therefore, yearly rent

= Rs.5000 ×

Now, the company has hired one of its walls for 3 months.

Thus, rent paid by the company for 3 months

= 1320 ×

= 110 × 5000 × 3

= Rs. 1650000

Therefore, rent paid by the company for 3 months = Rs. 1650000.

Q3 )
There is a slide in a park. One of its side walls has been painted in same colour with a message “KEEP THE PARK GREEN AND CLEAN” (see figure). If the sides of the wall are 15
m, 11 m and 6 m, find the area painted in colour.

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

The given figure formed a triangle whose sides are:

a = 15m,b = 11m, c = 6m

(Since, 2s = a + b + c)

Therefore, area painted in colour

=

(Since, Heron's formula [area =

=

=

=

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Let the sides of a triangle a = 18 cm, b = 10 cm and c

We have, perimeter = 42 cm

So, a + b + c = 42

By substituting the values,

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

Q5 ) Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Suppose that the sides in cm, are 12x, 17x and 25x.

Then, we know that

12x + 17x + 25x = 540

(Given, Perimeter of the triangle)

54x = 540

So, the sides of the triangle are 12 × 10cm, 17 × 10cm, 25 × 10cm

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

=

= 100 × 10 × 9

Therefore, area of the given triangle is 9000

Q6 ) An isosceles triangle has perimeter 30 cm arid each of the equal sides is 12 cm. Find the area of the triangle.

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Let

We have,

AB = AC = 12cm ...(Given)

Now, AB + AC + BC = 30

(Given, perimeter = 30cm)

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

Q1 )
A park, in the shape of a quadrilateral ABCD, has

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

In the quadrilateral ABCD,

we have, right angled triangle

We have,

(By Pythagoras theorem)

=

Also, In

AB = 9m, BD = 13m, DA = 8m

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

Since,

Area of

(Since, Area of triangle =

=

Area of quadrilateral ABCD

= Area of

Hence from (i) and (ii),

Area of quadrilateral ABCD =

Q2 ) Find the area of a quadrilateral ABCD in which AB = 3cm, BC = 4cm, CD = 4cm, DA = 5cm and AC = 5cm.

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

In the quadrilateral ABCD,

we have,

We have, AB = 3cm, BC = 4cm, CA = 5cm

We have,

(By Pythagoras theorem)

=

Hence,

Now in

AC = 5cm, CD = 4cm, DA = 5cm

We know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

Since,

Area of

(Since, Area of triangle =

=

Area of quadrilateral ABCD = Area of

Hence from (ii) and (iii),

Area of quadrilateral ABCD =

Q3 )
Radha made a picture of an aeroplane with coloured paper as shown in figure. Find the
total area of the paper used.

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

For part I :

It is a triangle with sides 5 cm, 5 cm and 1 cm.

Thus, We know that,

Now, Area of part I triangle

=

(Since, Heron's formula [area =

=

=

=

For part II :

It is a rectangle with sides 6.5 cm and 1 cm

(Since, Area of rectangle = Lenght × Breadth)

= 6.5

For part III :

It is a trapezium ABCD.

(Since, Area of triangle =

Now, Area of trapezium

=

=

=

=

= 3 × 0.433

= 1.299

Therefore, Area of part III is 1.299

For part IV & V :

Both the parts IV and V are the same.

Area of part IV & V =

So, we get,

Total area of paper used = Area of (I + II + III + IV + V)

= (2.487 + 6.5 + 1.299 + 4.5 + 4.5)

= 19.286

= 19.3

Q4 ) A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram?

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Let ABC be a triangle with sides AB = 26 cm, BC = 28 cm, CA = 30 cm

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

We know that,

Area of parallelogram = Base × Height ...(ii)

We also have,

Area of parallelogram = Area of

Thus, from (i) and (ii),

Therefore, the height of the parallelogram is 12cm.

Q5 ) A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Let ABCD be a rhombus.

Area of the rhombus ABCD

= 2 × area of

As we know that, in a rhombus, diagonals divides it in two equal parts.

In

AB = 30 m, BD = 48 m, DA=30 m

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

Thus, from Equation (i),

Area of rhombus

= 2 × 432m

= 864 m

Also, Number of cows = 18

So, therefore, Area of grass field per cow

=

Q6 )
An umbrella is made by stitching 10 triangular pieces of cloth of two different colours (see figure), each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Each triangular piece in an umbrella is an isosceles triangle with sides 50 cm, 50 cm, 20 cm.

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

Thus, from Equation (i),

Area of each triangular piece

= 2 × 432 cm = 864 cm

Since, there are 10 triangular pieces, out of which 5 & 5 are of different colours.

Hence, total area of cloth of each colour =

Q7 )
A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in figure. How much paper of each shade has been used in it?

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Since, the kite is in the shape of a square.

Each diagonal of square = 32 cm ...(Given)

We know that, the diagonals of a square bisect each other at right angle.

For Part I :

Area of Part I

=

=

= 16 × 16 = 256

For Part II :

Area of Part II

=

=

= 16 × 16 = 256

For Part III :

It is a triangle with sides 6 cm, 6 cm and 8 cm.

Now, we know that,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

= 8 × 2.24 cm = 17.92

Hence, the area of colour used for Paper I = 256

The area of colour used for Paper II = 256

The area of colour used for Paper III = 17.92

Q8 )
A floral design on a floor is made up of 16 tiles which are triangular, the sides of the triangle being 9 cm, 28 cm and 35 cm (see figure). Find the cost polishing the tiles at the rate of Rs.50 paise per

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Given,

the sides of a triangular tile are 9 cm, 28 cm and 35 cm.

For each triangular tile, we have,

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

= 16 ×

= 16 × 36 × 2.45

Total cost of ploshing the titles at the rate of Rs.50 paise per

= Rs.

Q9 ) A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non - parallel sides are 14 m and 13 m. Find the area of the field.

NCERT Solutions for Class 9 Maths Chapter 12 Herons Formula

Answer :

Here, ABCD is a trapezium and AB || DC.

Here, ABCD is a trapezium and CE || DA and

Now, AB = 25 m, BC = 14 m, CD = 10 m, DA = 13 m, AE = 10 m and CE = 13m

Now, for

Now, Area of triangle

=

(Since, Heron's formula [area =

=

=

=

=

Also, Area of triangle

=

=

Now, area of the trapezium ABCD

=

=

=

=

= 35 × 5.65

Therefore, the area of the field is 196