Download 30-60-90 Right Triangle

30-60-90 Right Triangle
A 30-60-90 right triangle (literally pronounced "thirty sixty ninety") is a special type of right
triangle where the three angles measure 30 degrees, 60 degrees, and 90 degrees. The
triangle is significant because the sides exist in an easy-to-remember ratio: 1:1sqrt(3):2. That
is to say, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square
root of 3 times the shorter leg. You might also remember it as "X, 2X, and X roots of 3",
which is how I remember it, but then you have to remember that 2X is actually the longest
side, not X roots of 3.
Which side is which? The side opposite the 30 degree angle will have the shortest length.
The side opposite the 60 degree angle will be sqrt(3) times as long, and the side opposite
the 90 degree angle will be twice as long.
For example, given the 30-60-90 triangle below, find the lengths of the missing sides:
Since this is a 30-60-90 right triangle, we know that the sides exist in the proportion
1:sqrt(3):2. The shortest side, 1, is opposite the 30 degree angle. Since side X is opposite
the 60 degree angle, we know that it is equal to 1*sqrt(3), or about 1.73. Finally, side Y is
opposite the right angle, and it is twice the shortest side, or 2.
Where does the formula come from?
Is this just another made-up math formula? No! This is just an application of basic
trigonometry. For the example above, we could have taken the sine of the left-most angle,
sin(30) = 1/2. Because sine gives us the ratio of opposite over hypotenuse, we would know
that the hypotenuse would have to be 2. Basically, the 30-60-90 triangle is easy to solve
because the sine and cosine of those angles are also very simple.
Need another example?
Use the same principles to solve for the unknown variables X and Y. The known side is 4,
and that is the longest side. Remember how the longest side is twice the shortest side for a
30-60-90 triangle? That means Y must be 2! The side opposite the 60 degree angle equals
the shorter side times the square root of 3. Therefore X equals 2*sqrt(3).
Summary
For a right triangle with angles measuring 30, 60 and 90 degrees, the sides will have lengths
in a ratio of 1:sqrt(3):2, as shown in this diagram: