A 30-60-90 triangle can be formed by cutting an equilateral triangle in half. This is one of two special types of right triangles (the other is a 45-45-90 triangle, which is half of a square).
What do you know about the sides of equilateral triangle? All 3 sides of an equilateral triangle have the same length.
Let's say you have an equilateral triangle and all three sides are 10 units long. If you cut the equilateral triangle in half, it forms a 30-60-90 triangle. If the hypotenuse of the triangle is 10, what is the length of the shorter leg of the 30-60-90 triangle?
Do you know how long the missing side will be? The bottom side has been cut in half to form the 30-60-90 triangle, so it must be half of 10: 5 units long.
Relationship Between the Short Leg and Hypotenuse
To summarize, the short leg of a 30-60-90 triangle is always 1/2 the length of the hypotenuse. You could also switch it around and say that the hypotenuse is always twice the length of the short leg.
Example 1 Find a.
Answer: It can be helpful to sketch in the rest of the equilateral triangle to make it easier to see the relationship between the short leg and the hypotenuse. The side across from the right angle is the hypotenuse (the 14) and we need to find the short leg (remember, the legs form the L to make the 90 degree angle). We know that the hypotenuse is 14. This means the the short leg (the a) is half of 14. a = 7.
Example 2 Find b.
Answer: It might help to sketch in the rest of the equilateral triangle to help you see the relationship between the two sides. The legs always form the 90 degree corner and the hypotenuse is always across from the right angle. This means we know the short leg is 4 and we need to find the hypotenuse. The hypotenuse is always twice as long as the short leg so we can just multiply 4 by 2 to get b. b = 8
Open the next tab to see how the long leg is related to the other sides.
Long Leg of a 30-60-90 Triangle
We've figured out the relationship between the short leg and hypotenuse of a 30-60-90 triangle, but what about the longer leg? How do we find that one? It's not quite as simple.
The Long Way: Use the Pythagorean Theorem
There are two different ways to find the length of the long leg. One way is to use the Pythagorean Theorem. In the example below, we have a right triangle that's missing a side. We can use the Pythagorean Theorem to find this missing side.
When you use the Pythagorean Theorem, it's important to label a, b, and c correctly. a and b are the legs that form the right angle (they make an L shape). c must be the hypotenuse (the side across from the right angle). In the triangle above, the 10 is the hypotenuse so we need to set c = 10.
We could leave the answer for the hypotenuse as the square root of 75 or round it and get a decimal answer. To show the shortcut relationship, we can simplify the square root of 75 instead. When you're simplifying a square root, look for factors of the number under the square root that are perfect squares. See if you can divide the number by 4, 9, 16, 25, etc. . In this case, 25 is a perfect square that is a factor of 75. This means we can rewrite the problem as the square root of 25 times the square root of 3. The square root of 25 is 5, so this simplifies to 5 times the square root of 3.
Now look at all three sides of the 30-60-90 triangle. Do you see a shortcut you could use to find the long leg?
The Shortcut
The long leg is the same as the short leg times the square root of 3. This relationship is true of every 30-60-90 triangle. So from now on, don't use the Pythagorean Theorem. Use the shortcut. If you know the short leg, just multiply it by the square root of 3 to find the long leg.
Example 1 Find c and d.
Answer: It might help to sketch out the rest of the equilateral triangle so you can visualize the relationships between the sides. Remember, the two legs form the letter L that makes the 90 degree angle. We've been given the shorter leg.
We know the hypotenuse is twice as large as the short leg. This gives us d = 9x2 = 18 To find the longer leg, we multiply the short let by the square root of 3. This means c is 9 times the square root of 3.
Example 2 Find e and f.
Answer: We've been given the hypotenuse of the 30-60-90 triangle. We know that the short leg is always half the length of the hypotenuse. Half of 26 is 13.
To find the long leg, we just multiply the short leg by the square root of 3.