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?

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

Now, we know that,

\(s = \frac{a + b + c}{2}\)
\(\therefore \) \(s = \frac{50 + 50 + 20}{2} = \frac{120}{2} = 60cm\)

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

= \(\sqrt{60 × 10 × 10 × 40}\)
= \(\sqrt{3 × 2 × 10 × 10 × 10 × 10 × 2 × 2}\)
= \(2 × 100\sqrt{6} {cm}^2\)
= \(200\sqrt{6} {cm}^2\)

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 = \(5 × 200\sqrt{6} {cm}^2\) = \(1000\sqrt{6} {cm}^2\)