# The length of the small leg of a 30°-60°-90° triangle is 3. What is its perimeter?

Nov 25, 2015

To compute the perimeter of a triangle, you need to know the length of all sides.

Let's call the small leg $a$, the big leg $b$ and the hypotenuse $c$.

We already know that $a = 3$. Now, let's compute the values of $b$ and $c$.

First, we can compute $b$ using the $\tan$:

$\tan = \left(\text{opposite")/("adjacent}\right)$

=> tan 60° = b/a = b / 3

=> b = tan 60° * 3 = sqrt(3) * 3

Now, we can compute $c$ either with one of the trigonometric functions or with theorem of Pythagoras:

${a}^{2} + {b}^{2} = {c}^{2}$

${3}^{2} + {\left(\sqrt{3} \cdot 3\right)}^{2} = {c}^{2}$
$\iff 9 + 27 = {c}^{2}$

$\iff c = 6$

Now that we have all three sides, we can compute

$P = a + b + c = 3 + 3 \sqrt{3} + 6 = 9 + 3 \sqrt{3} \approx 14.196$