NCERT Solutions for Class 9 Maths Unit 12

Heron's Formula Class 9

Unit 12 Heron's Formula Exercise 12.1, 12.2 Solutions

Introduction to Heron's Formula
Heron's formula for the area of a triangle with sides of length a, b, c is

A = sqrt(s (s-a) (s-b) (s-c))
where
s = (a+b+c)/2

It is unfortunate that this topic has essentially disappeared from school curriculum today. Calculation, given available calculations and computers, can no longer be a reason for avoiding the formula. In what follows, I hope to show some interesting and challenging problems using Heron's formula.
Whether or not one would pose the demonstration or proof of Heron's formula for a particular class would depend on the class. Initially, exploration with Heron's formula could involve computing areas using the formula and making comparison's of the results -- much as we pose analogous exercises in a meaningful way with the Pythagorean theorem long before a proof or demonstration is fully understood.

Exercise 12.1 : Solutions of Questions on Page Number : 202

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?

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

Q2 :

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 m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?

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

2s = 264 m

s = 132 m

By Heron's formula,

Rent of 1 m2 area per year = Rs 5000

Rent of 1 m2 area per month = Rs

Rent of 1320 m2 area for 3 months =

= Rs (5000 × 330) = Rs 1650000

Therefore, the company had to pay Rs 1650000.

Q3 :

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.

Q4 :

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

Q5 :

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

Q6 :

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

Exercise 12.2 : Solutions of Questions on Page Number : 206

Q1 :

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?

Q2 :

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.

Q3 :

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

Q4 :

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.

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?

Q6 :

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?

Q7 :

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?

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 the given figure). Find the cost of polishing the tiles at the rate of 50p per cm2.

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.