# NCERT Solutions for Class 9 Maths Chapter 12 – Heron’s Formula

Here we provide NCERT Solutions for Class 9 Maths Chapter 12 – Heron’s Formula for English medium students, Which will very helpful for every student in their exams. Students can download the latest NCERT Class 9 Maths solution Chapter 12 pdf. Now you will get step by step solution to each question.                                        = √15(15 – 12) (15 – 12) (15 – 6)cm2
= √15 × 3 × 3 × 9 cm2
= 9√15 cm2

Page No: 206

Exercise 12.2

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

∠C = 90º, AB = 9 m, BC = 12 m, CD = 5 m and AD = 8 m
BD is joined.

In ΔBCD,
By applying Pythagoras theorem,
BD2 = BC+ CD
⇒ BD2 = 12+ 5
⇒ BD2 = 169
⇒ BD = 13 m
Area of ΔBCD = 1/2 × 12 × 5 = 30 m2
Now,
Semi perimeter of ΔABD(s) = (8 + 9 + 13)/2 m = 30/2 m = 15 m
Using heron’s formula,
Area of ΔABD  = √s (s-a) (s-b) (s-c)
= √15(15 – 13) (15 – 9) (15 – 8) m2
= √15 × 2 × 6 × 7 m2
= 6√35 m2 = 35.5 m2 (approx)

Area of quadrilateral ABCD = Area of ΔBCD + Area of ΔABD = 30 m2 + 35.5m2 = 65.5m2

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.

AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm

In ΔABC,
By applying Pythagoras theorem,
AC2 = AB+ BC
⇒ 52 = 3+ 4
⇒ 25 = 25
Thus, ΔABC is a right angled at B.
Area of ΔBCD = 1/2 × 3 × 4 = 6 cm2
Now,
Semi perimeter of ΔACD(s) = (5 + 5 + 4)/2 cm = 14/2 cm = 7 m
Using heron’s formula,
Area of ΔABD  = √s (s-a) (s-b) (s-c)
= √7(7 – 5) (7 – 5) (7 – 4) cm2
= √7 × 2 × 2 × 3 cm2
= 2√21 cm= 9.17 cm2 (approx)

Area of quadrilateral ABCD = Area of ΔABC + Area of ΔABD = 6 cm+ 9.17 cm2 = 15.17 cm

3. Radha made a picture of an aeroplane with coloured paper as shown in Fig 12.15. Find the total area of the paper used.

Length of the sides of the triangle section I = 5cm, 1cm and 5cm
Perimeter of the triangle = 5 + 5 + 1 = 11cm
Semi perimeter = 11/2 cm = 5.5cm
Using heron’s formula,
Area of section I  = √s (s-a) (s-b) (s-c)
= √5.5(5.5 – 5) (5.5 – 5) (5.5 – 1) cm2
= √5.5 × 0.5 × 0.5 × 4.5 cm2
= 0.75√11 cm= 0.75 × 3.317cm2 = 2.488cm2 (approx)
Length of the sides of the rectangle of section I = 6.5cm and 1cm
Area of section II = 6.5 × 1 cm2 =  6.5 cm2
Section III is an isosceles trapezium which is divided into 3 equilateral of side 1cm each.
Area of the trapezium = 3 × √3/4 × 12 cm2 = 1.3 cm2 (approx)
Section IV and V are 2 congruent right angled triangles with base 6cm and height 1.5cm
Area of region IV and V = 2 × 1/2 × 6 × 1.5cm2 = 9cm2
Total area of the paper used = (2.488 + 6.5 + 1.3 + 9)cm2 = 19.3 cm2

4. 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.

Given,
Area of the parallelogram and triangle are equal.

Length of the sides of the triangle are 26 cm, 28 cm and 30 cm.
Perimeter of the triangle = 26 + 28 + 30 = 84 cm
Semi perimeter of the triangle = 84/2 cm = 42 cm
Using heron’s formula,
Area of the triangle = √s (s-a) (s-b) (s-c)
= √42(42 – 26) (46 – 28) (46 – 30) cm2
= √46 × 16 × 14 × 16 cm2
= 336 cm2Let height of parallelogram be h.
Area of parallelogram = Area of triangle
28cm × h = 336 cm2
h = 336/28 cm
⇒ h = 12 cm
The height of the parallelogram is 12 cm.

Page No: 207

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?

Diagonal AC divides the rhombus ABCD into two congruent triangles of equal area.

Semi perimeter of ΔABC = (30 + 30 + 48)/2 m = 54 m
Using heron’s formula,
Area of the ΔABC = √s (s-a) (s-b) (s-c)
= √54(54 – 30) (54 – 30) (54 – 48) m2
= √54 × 24 × 24 × 6 cm2
= 432 m2
Area of field = 2 × area of the ΔABC = (2 × 432)m= 864 m2
Thus,
Area of grass field which each cow will be getting = 864/18 m2 = 48 m2

6. An umbrella is made by stitching 10 triangular pieces of cloth of two different colours (see Fig.12.16), each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?

Semi perimeter of each triangular piece = (50 + 50 + 20)/2 cm = 120/2 cm = 60cm
Using heron’s formula,
Area of the triangular piece = √s (s-a) (s-b) (s-c)
= √60(60 – 50) (60 – 50) (60 – 20) cm2
= √60 × 10 × 10 × 40 cm2
= 200√6 cm2
Area of triangular piece = 5 × 200√6 cm= 1000√6 cm2

7. 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 Fig. 12.17. How much paper of each shade has been used in it?

We know that,
As the diagonals of a square bisect each other at right angle.
Area of given kite = 1/2 (diagonal)2
= 1/2 × 32 × 32 = 512 cm2
⇒ 512/2 cm= 256 cm2
So, area of paper required in each shade = 256 cm2
For the III section,
Length of the sides of triangle = 6cm, 6cm and 8cm
Semi perimeter of triangle = (6 + 6 + 8)/2 cm = 10cm
Using heron’s formula,
Area of the III triangular piece = √s (s-a) (s-b) (s-c)
= √10(10 – 6) (10 – 6) (10 – 8) cm2
= √10 × 4 × 4 × 2 cm2
= 8√6 cm2

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 Fig. 12.18). Find the cost of polishing the tiles at the rate of 50p per cm2 .

Semi perimeter of the each triangular shape = (28 + 9 + 35)/2 cm = 36 cm
Using heron’s formula,
Area of the each triangular shape = √s (s-a) (s-b) (s-c)
= √36(36 – 28) (36 – 9) (36 – 35) cm2
= √36 × 8 × 27 × 1 cm2
= 36√6 cm= 88.2 cm2
Total area of 16 tiles = 16 × 88.2 cm= 1411.2 cm2Cost of polishing tiles = 50p per cm2
Total cost of polishing the tiles = Rs. (1411.2 × 0.5) = Rs. 705.6

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.

Let ABCD be the given trapezium with parallel sides AB = 25m and CD = 10mand the non-parallel sides AD = 13m and BC = 14m.
CM ⊥ AB and CE || AD.
In ΔBCE,
BC = 14m, CE = AD = 13 m and
BE = AB – AE = 25 – 10 = 15m
Semi perimeter of the ΔBCE = (15 + 13 + 14)/2 m = 21 m
Using heron’s formula,
Area of the ΔBCE = √s (s-a) (s-b) (s-c)
= √21(21 – 14) (21 – 13) (21 – 15) m2
= √21 × 7 × 8 × 6 m2
= 84 m
also, area of the ΔBCE = 1/2 × BE × CM = 84 m
⇒ 1/2 × 15 × CM = 84 m
⇒ CM = 168/15 m
⇒ CM = 56/5 m
Area of the parallelogram AECD = Base × Altitude = AE × CM = 10 × 84/5 = 112 m
Area of the trapezium ABCD = Area of AECD + Area of ΔBCE
= (112+ 84) m2  = 196 m2

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