An umbrella is made by stitching $10$ triangular pieces of cloth of two different colours (see figure),each piece measuring $20\, cm, 50\, cm$ and $50\, cm$. How much cloth of each colour is required for the umbrella?

(N/A) The sides of each triangular piece are $a = 20\, cm$,$b = 50\, cm$,and $c = 50\, cm$.
The semi-perimeter $s$ is calculated as:
$s = \frac{a + b + c}{2} = \frac{20 + 50 + 50}{2}\, cm = \frac{120}{2}\, cm = 60\, cm$.
Using Heron's formula,the area of each triangular piece is:
Area $= \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{60(60 - 20)(60 - 50)(60 - 50)}\, cm^2$
$= \sqrt{60 \times 40 \times 10 \times 10}\, cm^2 = \sqrt{240000}\, cm^2 = 200\sqrt{6}\, cm^2$.
Since there are $10$ triangular pieces in total,there are $5$ pieces of each colour.
Area of cloth required for each colour $= 5 \times 200\sqrt{6}\, cm^2 = 1000\sqrt{6}\, cm^2$.