Heron, a mathematician gave a formula for finding the area of a triangle in terms of the three sides. The formula given by him is also known as Heron's Formula and is stated below:
If a, b, c denote the sides BC, AC and AB respectively of a triangle ABC, then
Area of triangle ABC
[tex] = \sqrt{s(s - a)(s - b)(s - c)} [/tex]where [tex]s = \frac{a + b + c}{2} [/tex]
,the semi-perimeter of ∆ABC
or 2s = a + b + c
Note: This formula is useful in finding the area of a triangle when it is not possible to find the area of the triangle easily.
Now, comes to your question,
Let the sides of the triangle be a = 19 m, b = 21m and c = 15 m
[tex] s = \frac{a + b + c}{2} = \frac{19 + 21 + 15}{2} = \frac{55}{2} = 27.50 \: m[/tex]
∴ Area of the triangle
[tex] = \sqrt{s(s - a)(s - b)(s - c)} [/tex]
[tex] = \sqrt{27.50(27.50 - 19)(27.50 - 21)(27.50 - 15)} [/tex]
[tex] = \sqrt{27.50 \times 8.50 \times 6.50\times 12.50} [/tex]
[tex] = \sqrt{233.75 \times 6.50 \times 12.50} [/tex]
[tex] = \sqrt{1519.38 \times 12.50} [/tex]
[tex] = \sqrt{18992.25}[/tex]
= 137.81
A => 137.81 = 138 square metres
