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 $x:x3 :2x$, 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 $3 $.

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

$a_{2}+b_{2}1_{2}+3 _{2}1+34 =c_{2}=2_{2}=4=4 $

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 $183 $.
What is the perimeter of the triangle?
A. $18+62 $
B. $363 $
C. $36$
D. $24$
E. $18+63 $

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

(spoiler)

Answer: E. $18+63 $

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 $180_{∘}$, and since we’re given the right angle of $90_{∘}$ and another angle of $60_{∘}$, the last unknown angle must be $30_{∘}$.

The sides of any 30-60-90 triangle have a ratio of $x:x3 :2x$, 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=x3 $.

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