# I do not understand where Po shen loh got the 3√3. How did he get that?

• Module 2 Week 1 Day 3 Challenge Part 3

I do not understand where Po shen loh got the 3√3 for the bottom length of the triangle. How did he get that?

• @mirthfulostrich Thank you for asking! This is a very special right triangle called a $$30^{\circ}-60^{\circ}-90^{\circ}$$ right triangle, since it has angles of $$30^{\circ}, 60^{\circ}, \text{ and } 90^{\circ}.$$ It is one of the most popular triangles.

After finishing Module 2, you will have probably seen this triangle $$50$$ times! If you participate in any math competition at the middle-school or high-school level, you will probably see this triangle pop up twice on any exam! Prof. Loh has probably done $$2,000$$ math problems in his lifetime so far which involve this special $$30^{\circ}-60^{\circ}-90^{\circ}$$ right triangle! Consequently, he has memorized the ratios of its lengths, and in not too long, you will, too!

This triangle first made its appearance in the Day 1 Challenge Part 2 lesson, when we were finding the area of a regular hexagon. Someone has already asked about how to find the side lengths of a $$30^{\circ}-60^{\circ}-90^{\circ}$$ right triangle, so please visit this post to learn about how to derive the ratios of the sides of the triangle!

The basic idea is this:

$$\textcolor{red}{\text{ Take half of an equilateral triangle. This gives us two of the sides (the short leg and the hypotenuse).}}$$
$$\textcolor{red}{\text{ Then, use the Pythagorean Formula to find the long leg. }}$$ 