Answer :

Each triangular piece in an umbrella is an isosceles triangle with sides 50 cm, 50 cm, 20 cm.

Now, we know that,

$s=\frac{a+b+c}{2}$
$?$ $s=\frac{50+50+20}{2}=\frac{120}{2}=60cm$

Now, Area of triangle
= $\sqrt{60(60?50)(60?50)(60?20)}$
(Since, Heron's formula [area = $\sqrt{s(s?a)(s?b)(s?c)}$])

= $\sqrt{60\times 10\times 10\times 40}$
= $\sqrt{3\times 2\times 10\times 10\times 10\times 10\times 2\times 2}$
= $2\times 100\sqrt{6}{cm}^2$
= $200\sqrt{6}{cm}^2$

Thus, from Equation (i),

Area of each triangular piece
= 2 × 432 cm = 864 cm

Since, there are 10 triangular pieces, out of which 5 & 5 are of different colours.

Hence, total area of cloth of each colour = $5\times 200\sqrt{6}{cm}^2$ = $1000\sqrt{6}{cm}^2$