# Herons Formula

## Class 9 NCERT Maths

### NCERT

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?

Side of traffic signal board =$a$$\\ Perimeter of traffic signal board = 3 * a$$\\$ $2s=3a \Rightarrow s=\dfrac{3}{2}a$$\\ By Heron’s formula, Area of triangle =\\ \sqrt{s(s-a)(s-b)(s-c)}$$\\$ Area of given triangle=$\\$ $\sqrt{\dfrac{3}{2}a(\dfrac{3}{2}a-a)(\dfrac{3}{2}a-a)(\dfrac{3}{2}a-a)}$$\\ =\sqrt{\dfrac{3}{2}a(\dfrac{a}{2})(\dfrac{a}{2})(\dfrac{a}{2})}$$\\$ $=\dfrac{\sqrt{3}}{2}a^2....(1)$$\\ Perimeter of traffic signal board = 180 cm$$\\$ Side of traffic signal board $(a) =(\dfrac{180}{3}) cm =60 cm$$\\ Using Equation (1), area of traffic signal board =\dfrac{\sqrt{3}}{2}(60 cm)^2$$\\$ $=(\dfrac{3600}{4}\sqrt{3}) cm^2$$\\ =900\sqrt{3}cm^2 2 The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m, and 120 m (see the given figure). The advertisements yield an earning of Rs. 5000 per m^2 per year. A company hired one of its walls for 3 months. How much rent did it pay? ##### Solution : The sides of the triangle (i.e., a, b, c) are of 122 m, 22 m, and 120 m respectively.\\ Perimeter of triangle = (122 + 22 + 120) m$$\\$ $2s = 264 m$$\\ s = 132 m$$\\$ By Heron’s formula,$\\$ Area of triangle =$\sqrt{s(s-a)(s-b)(s-c)}$$\\ Area of given triangle =[\sqrt{132(132-122)(132-22)(132-120)}]m^2$$\\$ $=[\sqrt{132(10)(110)(12)}]$$\\ Rent of 1 m^2 area per year = Rs. 5000$$\\$ Rent of $1 m^2$ area per month = Rs.$(\dfrac{5000}{15}*3*1320)$$\\ Rs.(5000*330)=Rs 1650000$$\\$ Therefore, the company had to pay Rs. $1650000.$

3   The floor of a rectangular hall has a perimeter $250 m$. If the cost of panting the four walls at the rate of Rs. $10$ per $m^2$ is Rs.$15000,$ find the height of the hall. [Hint: Area of the four walls = Lateral surface area.]

##### Solution :

Let length, breadth, and height of the rectangular hall be $l$ m, $b$ m, and $h$ m respectively.$\\$ Area of four walls = $2lh + 2bh = 2(l + b) h$$\\ Perimeter of the floor of hall = 2(l + b) = 250 m$$\\$ $\therefore$ Area of four walls = $2(l + b) h = 250h m^2$$\\ Cost of painting per m^2 area = Rs. 10$$\\$ Cost of painting $250h m^2$ area = Rs.$(250h × 10) = Rs. 2500h$$\\ However, it is given that the cost of paining the walls is Rs. 15000.$$\\$ $\therefore 15000 = 2500h$$\\ h = 6$$\\$ Therefore, the height of the hall is $6 m.$

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