If the height of an equilateral triangle is 20$$\surd2$$ cm, then what is its area (in cm$$^2$$)?

The height of an equilateral triangle is 20$$\surd2$$ cm.

height = h = 20$$\surd2$$ cm

As per Pythagoras theorem, $$a^2\ =\ \left(\frac{a}{2}\right)^2\ +\ h^2$$

$$a^2\ =\ \frac{a^2}{4}\ +\ \left(20\surd2\right)^2$$

$$\frac{3a^2}{4} = 800$$

$$a^2 = \frac{3200}{3}$$

a = $$\frac{40\sqrt{\ 2}}{\sqrt{\ 3}}$$

area of equilateral triangle = $$\frac{1}{2}\times\ a \times\ h$$

$$=\frac{1}{2}\times\ \frac{40\sqrt{\ 2}}{\sqrt{\ 3}}\times\ 20\surd2$$

$$=\frac{1}{2}\times\ \frac{40}{\sqrt{\ 3}}\times\ 40$$

$$=\frac{20}{\sqrt{\ 3}}\times\ 40$$
$$=\frac{800}{\sqrt{\ 3}} $$
Multiply by $$\sqrt{\ 3}$$ in numerator and denominator.

$$=\frac{800}{\sqrt{\ 3}}\times\ \frac{\sqrt{\ 3}}{\sqrt{\ 3}}$$
$$=\frac{800}{3}\sqrt{3}$$