Chapter 12: Heron’s Formula

EXERCISE 12.1

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 ?

Solution:  Since a is the side of an equilateral triangle .

 So, the perimeter of an equilateral triangle is 2s .

The semi-perimeter of an equilateral triangle

s=a+a+a2=3a2

The area of the signal board =ss-as-as-a

=3a2 3a2-a3a2-a3a2-a

=3a23a-2a23a-2a23a-2a2

=3a2×a2×a2×a2

=3a416=34a²

Second part : Given, the perimeter =180 

⇒a+a+a=180

⇒3a=180

⇒a=1803

⇒a=60 

The area of the signal board

=34a2=34×60×60 cm2

=3×30×30 cm2

=9003 cm² 

2. The triangular side walls of a flyover have been used for advertisements . The sides of the walls are 122 m , 22 m and 120 m (see Fig. 12.9) . The advertisements yield an earning of Rs 5000 per m2 per year . A company hired one of its walls for 3 months . How much rent did it pay ?

Solution : Here , a=122 m , b=22 m , c=120 m

s=a+b+c2

=122+22+1202

=2642=132 m

 s-a=132-122=10

 s-b=132-22=110

 s-c=132-120=12 

The area of the one triangular wall =ss-as-bs-c

 =132×10×110×12

 =4×3×11×10×10×11×4×3

=4×3×10×11=1320 m² 

The area of two triangular wall =2×1320 m2

                                                        =2640 m² 

The rent pay by company

=Rs5000m2×2640 m2×312

=Rs  5000×2640×14

=Rs 5000×660 

=Rs 3300000 

3. There is a slide in a park . One of its side walls has been painted in some colour with a message ‘‘ KEEP THE PARK GREEN AND CLEAN ’’ (see Fig. 12.10) . If the sides of the wall are 15 m , 11 m and 6 m , find the area pointed in colour .

Solution : Here , a=15 m , b=11 m  and c=6 m 

s=a+b+c2

=15+11+62=322=16 m

s-a=16-15=1 m 

s-b=16-11=5 m 

s-c=16-6=10 m 

The area of the triangle wall =ss-as-bs-c

=16×1×5×10 m2

=4²×1×5×2×5 m²

=4×52 m2

=202 m²

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

Solution:  let c be the third side of the triangle .

Here, =18 cm  ,  b=10 cm  and  2s=42 cm 

⇒s=422

⇒s=21 cm

∴ a+b+c=42

⇒18+10+c=42

⇒c+28=42

⇒c=42-28

⇒c=14

s-a=21-18=3 cm 

 s-b=21-10=11 cm

 s-c=21-14=7 cm 

 The area of the triangle =ss-as-bs-c

=21×3×11×7 cm2

=3×7×3×11×7 cm2

=3×711cm2

=2111cm²

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

Solution : Let 12x , 17x and 25x are the sides of the triangle .

 Here,  a=12x , b=17x , c=25x and  2s=540

⇒s=5402

⇒s=270 cm

2s=540

⇒12x+17x+25x=540

⇒54x=540

⇒x=54054

⇒x=10

a=12x=12×10=120 cm , b=17x=17×10=170 cm and  c=25x=25×10=250 cm 

s-a=270-120=150 cm

s-b=270-170=100 cm

s-c=270-250=20 cm

The area of the triangle =ss-as-bs-c

=270×150×100×20 cm2

=3×9×10×3×5×10×10×10×4×5  cm²

=32×32×102×52×102×22 cm2

=3×3×10×5×10×2 cm2

=9000 cm²

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

Solution : let c be the third side of an isosceles triangle .

2s=30

⇒s=3015

⇒s=15 cm 

Again , 12+12+c=30

 ⇒24+c=30

 ⇒c=30-24

 ⇒c=6 cm 

Here , a=12 cm ,  b=12 cm , c=6 cm  and s=15 cm 

s-a=15-12=3 cm 

 s-b=15-12=3 cm 

 s-c=15-6=9 cm 

The area of the triangle =ss-as-bs-c

=15×3×3×9  cm2

=15×32×32 cm2

=3×315 cm2

=915 cm²

EXERCISE 12.2

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?

Solution :
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 .

Solution :
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.    Fig. 12.15
Solution :

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.

Solution:

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?
Solution:

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?
Solution :

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?  Fig. 12.16 Fig. 12.17
Solution :

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 .
Solution:

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.

Solution: