What Are 30°–60°–90° Triangles?

Since this is a “$30$${}^{\circ }$-$60$${}^{\circ }$-$90$${}^{\circ }$” triangle, you already know that the shorter leg measures half the length of the hypotenuse, or $k=\frac{12\text{cm}}{2}=6\text{cm}$. You can now find the lengtt of the longer leg either by using the Pythagorean theorem, or by taking $\sqrt{3}k$. Here I’ll show you both of them.

First, the Pythagorean theorem:

With $\sqrt{3}k$:

The longer leg is about $10.39$ cm long.

Since this is a “$30$${}^{\circ }$-$60$${}^{\circ }$-$90$${}^{\circ }$” triangle, you know that the shorter leg’s length is half of the hypotenuse $h$. Therefore you can make the following equation using the Pythagorean theorem:

The hypotenuse measures about $3.5$ cm. The shorter leg is half of the hypotenuse, so it is

Because the angles in an equilateral triangle are all $60$${}^{\circ }$, you can find the height of the triangle by using the formula for a “$30$${}^{\circ }$-$60$${}^{\circ }$-$90$${}^{\circ }$” triangle. The height becomes one of the legs in a “$30$${}^{\circ }$-$60$${}^{\circ }$-$90$${}^{\circ }$” triangle:

As you can see, the hypotenuse is $6$ cm, and the shorter leg of the new triangle is exactly half of the hypotenuse (and half of the length of the side of the original triangle), which is $3$ cm.

You then calculate:

The height $h$ is therefore about $5.2$ cm.