# Ex.12.1 Q3 Heron’s Formula Solution - NCERT Maths Class 9

## Question

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 Figure. If the sides of the wall are \(15 \)m, \(11 \)m and \(6 \)m, find the area painted in colour.

## Text Solution

**What is known?**

Sides of the wall i.e. Dimensions of the triangle.

**What is unknown?**

Area of the (triangle) i.e. area of slope painted in colour.

**Reasoning:**

By using Heron’s formula we can calculate the area of triangle.

The formula given by Heron about the area of a triangle

\(=\sqrt{s(s-a)(s-b)(s-c)}\)

Where \(a, b\) and \(c\) are the sides of the triangle, and

\[\begin{align}s &= \text{Semi-perimeter}\\& = \begin{Bmatrix} \text{Half the Perimeter } \\ \text{ of the triangle} \;\end{Bmatrix} \\&=\frac{(a+b+c)}{2}\end{align}\]

**Steps:**

The sides of the walls (triangle) are \(a =11\; {\rm{m}} , b = 6 \, {\rm{m}}\;\text{and} \; c=15\, {\rm{m}}. \)

Semi Perimeter:

\(\begin{align}s & =\frac{(a+b+c)}{2} \\ & =\frac{11+6+15}{2} \\ &=\frac{32}{2} \\ &=16 \mathrm{m}\end{align}\)

By using Heron’s formula,

Area of a triangle

\(=\sqrt{s(s-a)(s-b)(s-c)}\)

Area of a triangle wall:

\(\begin{align}&=\sqrt{s(s-a)(s-b)(s-c)} \\ &=\!\sqrt{\!16(16\!-\!11)(16\!-\!6) (16\!-\!15) } \\ &=\sqrt{16 \times 5 \times 10 \times 1} \\ &=\sqrt{800} \mathrm{m}^{2} \\ &=20 \sqrt{2} \mathrm{m}^{2}\end{align}\)

Area of the wall of park painted in color \( = 20 \sqrt{2} \;\rm{m^2}\)