**Heron's Formula-**Students learned about figures

**of different shapes such as squares, rectangles, triangles, and quadrilaterals in lower grades and are familiar with the calculation of the**

**perimeters**and

**areas**of

**figures**.

**Area**of the

**triangle**is already known using the

**base and height of the triangle**. Some more concepts related to the

**area of the triangles**will be taught in this chapter

**Heron's Formula**. We can apply the previously learned formula to the

**right-angled triangle**. But to calculate the

**area**of the

**equilateral triangle, isosceles triangle,**and

**scalene triangle**, we need

**Heron's formula.**

Starting with the topic-

**Area of a triangle- by Heron's formula**. 3 solved examples are given to make the application of formula clear to students. This particular topic is accompanied by interactive as well as interesting activities. Exercise 12.1 contains 6 questions, some are in the form of word problems in which students have to find the area of the triangle formed in different given situations.

After that

**Application of Heron's Formula in Finding Area of Quadrilaterals**is given.

- The
**area**of a**quadrilateral**whose**sides**and one**diagonal**are given can be calculated by dividing the**quadrilateral**into**triangles**and using**Heron's formula.**

**Application of Heron's Formula in Finding Area of Quadrilaterals**. Thus, the chapter contains the basic formula of

**Heron**to find the area of any triangle. This topic is further extended to finding the

**area of a quadrilateral**by dividing the

**quadrilateral**into

**triangles**.

To end the chapter, a summary of the chapter-

**Heron's Formula**is given.

#### Page No 202:

#### Question 1:

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?

#### Answer:

Side of traffic signal board = *a*

Perimeter of traffic signal board = 3 × *a*

By Heron’s formula,

Perimeter of traffic signal board = 180 cm

Side of traffic signal board

Using equation (1), area of traffic signal board

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#### Page No 202:

#### Question 2:

The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122m, 22m, and 120m (see the given figure). The advertisements yield an earning of Rs 5000 per m^{2} per year. A company hired one of its walls for 3 months. How much rent did it pay?

#### Answer:

The sides of the triangle (i.e., *a*, *b*, *c*) are of 122 m, 22 m, and 120 m respectively.

Perimeter of triangle = (122 + 22 + 120) m

2*s* = 264 m

*s* = 132 m

By Heron’s formula,

Rent of 1 m^{2} area per year = Rs 5000

Rent of 1 m^{2} area per month = Rs

Rent of 1320 m^{2} area for 3 months =

= Rs (5000 × 330) = Rs 1650000

Therefore, the company had to pay Rs 1650000.

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#### Page No 203:

#### Question 3:

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

#### Answer:

It can be observed that the area to be painted in colour is a triangle, having its sides as 11 m, 6 m, and 15 m.

Perimeter of such a triangle = (11 + 6 + 15) m

2 *s* = 32 m

*s* = 16 m

By Heron’s formula,

Therefore, the area painted in colour is.

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#### Page No 203:

#### Question 4:

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.

#### Answer:

Let the third side of the triangle be *x*.

Perimeter of the given triangle = 42 cm

18 cm + 10 cm + *x *= 42

*x* = 14 cm

By Heron’s formula,

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#### Page No 203:

#### Question 5:

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

#### Answer:

Let the common ratio between the sides of the given triangle be *x*.

Therefore, the side of the triangle will be 12*x*, 17*x*,
and 25*x*.

Perimeter of this triangle = 540 cm

12*x* + 17*x* + 25*x* = 540 cm

54*x* = 540 cm

*x* = 10 cm

Sides of the triangle will be 120 cm, 170 cm, and 250 cm.

By Heron’s formula,

Therefore, the area of
this triangle is 9000 cm^{2}.

#### Page No 203:

#### Question 6:

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

#### Answer:

Let the third side of this triangle be *x*.

Perimeter of triangle = 30 cm

12 cm + 12 cm + *x* = 30 cm

*x* = 6 cm

By Heron’s formula,

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#### Page No 206:

#### Question 1:

A park, in the shape of a quadrilateral ABCD, has ∠C = 90°, AB = 9 m, BC = 12 m, CD = 5 m and AD = 8 m. How much area does it occupy?

#### Answer:

Let us join BD.

In ΔBCD, applying Pythagoras theorem,

BD^{2} = BC^{2} + CD^{2}

= (12)^{2} + (5)^{2}

= 144 + 25

BD^{2} = 169

BD = 13 m

Area of ΔBCD

For ΔABD,

By Heron’s formula,

Area of triangle

Area of ΔABD

Area of the park = Area of ΔABD + Area of ΔBCD

= 35.496 + 30 m^{2 }= 65.496 m^{2} = 65.5 m^{2} (approximately)

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#### Page No 206:

#### Question 2:

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

#### Answer:

For ΔABC,

AC^{2} = AB^{2} + BC^{2}

(5)^{2} = (3)^{2} + (4)^{2}

Therefore, ΔABC is a right-angled triangle, right-angled at point B.

Area of ΔABC

For ΔADC,

Perimeter = 2*s* = AC + CD + DA = (5 + 4 + 5) cm = 14 cm

*s* = 7 cm

By Heron’s formula,

Area of triangle

Area of ABCD = Area of ΔABC + Area of ΔACD

= (6 + 9.166) cm^{2} = 15.166 cm^{2} = 15.2 cm^{2} (approximately)

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#### Page No 206:

#### Question 3:

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

#### Answer:

**For triangle I**

This triangle is an isosceles triangle.

Perimeter = 2*s* = (5 + 5 + 1) cm = 11cm

Area of the triangle

**For quadrilateral II**

This quadrilateral is a rectangle.

Area = *l × b* = (6.5 × 1) cm^{2 }= 6.5 cm^{2}

**For quadrilateral III**

This quadrilateral is a trapezium.

Perpendicular height of parallelogram

Area = Area of parallelogram + Area of equilateral triangle

= 0.866 + 0.433 = 1.299 cm^{2}

Area of triangle (IV) = Area of triangle in (V)

Total area of the paper used = 2.488 + 6.5 + 1.299 + 4.5 × 2

= 19.287 cm^{2}

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#### Page No 206:

#### Question 4:

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

#### Answer:

**For triangle**

Perimeter of triangle = (26 + 28 + 30) cm = 84 cm

2*s* = 84 cm

*s* = 42 cm

By Heron’s formula,

Area of triangle

Area of triangle

= 336 cm^{2}

Let the height of the parallelogram be *h*.

Area of parallelogram = Area of triangle

*h* × 28 cm = 336 cm^{2}

*h* = 12 cm

Therefore, the height of the parallelogram is 12 cm.

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#### Page No 207:

#### Question 5:

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?

#### Answer:

Let ABCD be a rhombus-shaped field.

For ΔBCD,

Semi-perimeter, = 54 m

By Heron’s formula,

Area of triangle

Therefore, area of ΔBCD

Area of field = 2 × Area of ΔBCD

= (2 × 432) m^{2} = 864 m^{2}

Area for grazing for 1 cow = 48 m^{2}

Each cow will get 48 m^{2} area of grass field.

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#### Page No 207:

#### Question 6:

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

#### Answer:

For each triangular piece,

Semi-perimeter,

By Heron’s formula,

Area of triangle

Since there are 5 triangular pieces made of two different coloured cloths,

Area of each cloth required

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#### Page No 207:

#### Question 7:

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

#### Answer:

We know that

Area of square (diagonal)^{2}

Area of the given kite

Area of 1^{st} shade = Area of 2^{nd} shade

Therefore, the area of paper required in each shape is 256 cm^{2}.

**For III**^{rd}** triangle**

Semi-perimeter,

By Heron’s formula,

Area of triangle

Area of III^{rd} triangle

Area of paper required for III^{rd} shade = 17.92 cm^{2}

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#### Page No 207:

#### Question 8:

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 the given figure). Find the cost of polishing the tiles at the rate of 50p per cm^{2}.

#### Answer:

It can be observed that

Semi-perimeter of each triangular-shaped tile,

By Heron’s formula,

Area of triangle

Area of each tile

= (36 × 2.45) cm^{2}

= 88.2 cm^{2}

Area of 16 tiles = (16 × 88.2) cm^{2}= 1411.2 cm^{2}

Cost of polishing per cm^{2} area = 50 p

Cost of polishing 1411.2 cm^{2} area = Rs (1411.2 × 0.50) = Rs 705.60

Therefore, it will cost Rs 705.60 while polishing all the tiles.

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#### Page No 207:

#### Question 9:

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.

#### Answer:

Draw a line BE parallel to AD and draw a perpendicular BF on CD.

It can be observed that ABED is a parallelogram.

BE = AD = 13 m

ED = AB = 10 m

EC = 25 − ED = 15 m

For ΔBEC,

Semi-perimeter,

By Heron’s formula,

Area of triangle

Area of ΔBEC

m^{2}= 84 m^{2}

Area of ΔBEC

$\Rightarrow 84=\frac{1}{2}\times 15\times \mathrm{BF}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{BF}=\frac{168}{15}=11.2\mathrm{m}$

Area of ABED = BF × DE = 11.2 × 10 = 112 m^{2}

Area of the field = 84 + 112 = 196 m^{2}

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