Find the height of an equilateral triangle of side 12 cm.

Let ABC be the equilateral triangle with AD as an altitude from A meeting BC at D. Then, D will be the midpoint of BC.
Applying Pythagoras theorem in right-angled triangle ABD, we get:



$$\begin{gathered} A{B}^2 = A{D}^2 + B{D}^2 \\ \Rightarrow A{D}^2 = {12}^2 - 6^2 (\because BD\hspace{0.17em}= \frac{1}{2}BC = 6) \\ \Rightarrow A{D}^2 = 144 - 36 = 108 \\ \Rightarrow AD = \sqrt{108} = 6\sqrt{3} \mathrm{cm} \end{gathered}$$

Hence, the height of the given triangle is 6$\sqrt{3}$ cm.