Right triangles (30-60-90)

Now that you’ve learned about 45-45-90 triangles, it won’t be surprising that a triangle with angle measures of 30°, 60°, and 90° is called a 30-60-90 triangle. These triangles look like this:

The side lengths of a 30-60-90 triangle always have the ratio $x:x\sqrt{3}:2x$. This ratio stays the same no matter how large or small the triangle is.

• The hypotenuse is always double the shortest side.
• The longer leg is always the shortest side multiplied by $\sqrt{3}$.

You can also derive these values using the Pythagorean theorem. For example, suppose the shortest side is $x=1$:

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

Example 30-60-90 triangle question

Now that we know the key facts, let’s try an exam-style 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. $18+6\sqrt{3}$

Try solving it on your own, then check your answer.

The figure gives enough information to guarantee this is a 30-60-90 triangle. The interior angles of any triangle add to $180^{\circ }$. Since one angle is $90^{\circ }$ and another is $60^{\circ }$, the remaining angle must be $30^{\circ }$.

In any 30-60-90 triangle, the side lengths have the ratio $x:x\sqrt{3}:2x$. Also, the area of a triangle is $A=bh/2$. In this diagram, you can match the base and height to the legs of the right triangle:

• base $b=x$
• height $h=x\sqrt{3}$

Now plug into the area formula and solve for $x$:

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

Now that you know the short side is $x=6$, use the ratio $x:x\sqrt{3}:2x$ to get all three side lengths: $6:6\sqrt{3}:12$.

Finally, add the side lengths to find the perimeter:

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

Memorize these special triangle ratios, and you’ll be able to recognize and solve many triangle questions quickly.