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

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Answer 1 · 2015-11-25T14:23:32

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 = ("opposite")/("adjacent")#

#=> 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 + (sqrt(3)*3)^2 = c^2#
# <=> 9 + 27 = c^2#

# <=> 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) ~~ 14.196#