Right triangles (30-60-90)

Now that you’ve learned about 45-45-90 triangles, it should be no surprise that a triangle with angle measurements of 30, 60, and 90 degrees makes… a 30-60-90 triangle. These triangles look like this:

The ratio of the side lengths of a 30-60-90 triangle is always $x:x\sqrt{3}:2x$, and this is always consistent regardless of the size of the triangle.

The hypotenuse of a 30-60-90 triangle is always double the smallest side length, and the third side is the smallest side length multiplied by $\sqrt{3}$.

These values can also be derived from the Pythagorean theorem. Imagine if the $x$ value in this triangle was $1$:

$$\begin{array}{rl}{a}^2+b^2& =c^2\\ 1^2+{\sqrt{3}}^2& =2^2\\ 1+3& =4\\ 4& =4\end{array}$$

Example 30-60-90 triangle question

Now that we know the facts, let’s try an exam question!

The triangle below has an area of $18\sqrt{3}$.
What is the perimeter of the triangle? A. $18+6\sqrt{2}$
B. $36\sqrt{3}$
C. $36$
D. $24$
E. $18+6\sqrt{3}$

See if you can solve it yourself, then check your answer!

The figure gives us enough information to guarantee that the shape is a 30-60-90 triangle. For every triangle, the sum of the interior angles is $180^{\circ }$, and since we’re given the right angle of $90^{\circ }$ and another angle of $60^{\circ }$, the last unknown angle must be $30^{\circ }$.

The sides of any 30-60-90 triangle have a ratio of $x:x\sqrt{3}:2x$, and the area of a triangle is $A=bh/2$. Matching this up with our 30-60-90 triangle, our base $b=x$, and the height $h=x\sqrt{3}$.

We can plug these variables into the area equation and solve for the length of the side $x$.

$$\begin{array}{rl}\text{Area}& =bh/2\\ 18\sqrt{3}& =x(x\sqrt{3})/2\\ 36\sqrt{3}& =x(x\sqrt{3})\\ 36& =x(x)\\ 36& =x^2\\ 6& =x\end{array}$$

Now that we know the short side $x=6$, we can apply the ratio of $x:x\sqrt{3}:2x$ to get all three sides of $6:6\sqrt{3}:12$.

The final step is to calculate the perimeter by adding up the sides.

$$\begin{array}{rl}P& =6+6\sqrt{3}+12\\ & =18+6\sqrt{3}\end{array}$$

Memorize these special triangle ratios and you’ll be able to solve triangle questions with ease!