Pointed, blunt, isosceles, equilateral …. When it comes to triangles, there are many different varieties, but only a few that are "special". These special triangles have sides and angles that are consistent and predictable and can be used to shortcut your way through your geometry or trigonometry problems. And a 30-60-90 triangle - pronounced "thirty sixty ninety" - is actually a very special kind of triangle.
In this guide, we'll explain what a 30-60-90 triangle is, why it works, and when (and how) you can use your knowledge of it. So let's tackle it!
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle (a right triangle is any triangle that contains a 90-degree angle) that always has angles of 30 degrees, 60 degrees, and 90 degrees. Since it is a special triangle, it also has side length values that are always in a constant ratio to each other.
These relationships are:
Side opposite the 30° angle: x
Side opposite the 60° angle: x * √3
Side opposite the 90° angle: 2x
For example, a 30-60-90 degree triangle might have the following side lengths:
2, 2√3, 4
7, 7√3, 14
√3,3, 2√3
(Why is the longer leg 3? Because in this triangle the shortest leg (x) is √3 and the longer leg is x√3 => √3 *√3 = √9 => 3)
Etc.
The side opposite the 30° angle is always the smallest, because 30 degrees is the smallest angle. The side opposite the 60° angle is the mean longitude, since 60 degrees is the mean angle in this triangle. Finally, the side opposite the 90° angle is always the largest side because 90 degrees is the largest angle.
Although it looks similar to other types of right triangles, a 30-60-90 triangle is special because you only need three pieces of information to find the rest of the measurements: two angle measurements and a side length (it doesn't matter which side).
For example, since they are all 30-60-90 triangles, we can fill in all the remaining information gaps of the triangles below.
We can see that this is a right triangle in which the hypotenuse is twice the length of one of the legs. This means this must be a 30-60-90 triangle and the specified leg is opposite the 30°. The longer leg must therefore face the 60° angle and measure 6 *√3 or 6√3.
We can see that this must be a 30-60-90 triangle because we are told that this is a right triangle with a given measure, 30°. The unmarked angle must then be 60°. Since 18 is the measure opposite the 60° angle, it must equal x√3. The shortest leg must then measure $18/√3$. And the hypotenuse is $2(18/√3)$.
Again, we are given two angle measurements (90° and 60°), so the third measurement is 30°. Since this is a 30-60-90 triangle and the hypotenuse is 30, the shortest leg is 15 and the longer leg is 15√3.
No need to consult the Magic Eight Ball - these rules always work.
Why It Works (30-60-90 Triangle Theorem Proof)
But why does this particular triangle work the way it does? How do we know these rules are legitimate? Let's go through exactly how the 30-60-90 theorem works and prove why these side lengths will always be consistent.
First, let's forget right triangles for a second and look at an equilateral triangle.
An equilateral triangle is a triangle that has all equal sides and all equal angles. Since the interior angles of a triangle always add up to 180° and €180/3 = €60, an equilateral triangle always has three 60° angles.
Now let's drop an elevation from the top corner to the base of the triangle.
We have now created two right angles and two congruent (equal) triangles. How do we know they are the same? Because we fell from a heightequilateralTriangle, we exactly divided the base in half. The new triangles also share a side length (the height) and they each have the same hypotenuse length. Since they have three side lengths in common, they are congruent.
[Note: The two triangles are not only congruent based on the principles of side-side-side-lengths, or SSS, but also based on side-angle-side (SAS), angle-angle-side (AAS), and Angle -Side Angle (ASA). Basically? They are definitely congruent.]
Now that we have proved the congruence of the two new triangles, we can see that the apex angles must each equal 30° degrees (because each triangle already has angles of 90° and 60°). This means we made two 30-60-90 triangles.
And because we know that we've bisected the base of the equilateral triangle, we can see that the side opposite the 30° angle (the shortest side) of each of our 30-60-90 triangles is exactly half the length of the hypotenuse . So let's call our original side lengthXand our length halved$x/2$.
Now all we have to do is find our median length that the two triangles have in common. To do this, we can simply use the Pythagorean theorem.
$(x/2)^2 + b^2 = x^2$
$b^2 = x^2 – ({x^2}/4)$
$b^2 = 4x^2 – {x^2}/4$
$b^2 = {3x^2}/4$
$b = {√3x}/2$
So we are left with: $x/2, {x√3}/2, x$
Now let's multiply each bar by 2 just to make life easier and avoid all the breaks. So we are left with:
x, x√3, 2x
We can therefore see that the side lengths for a 30-60-90 triangle becomealwayshave consistent side lengths ofx, x√3, And2x(or $x/2,√3x/2$ andX).
Fortunately, we can prove the 30-60-90 triangle rules without all of that.
When to use the 30-60-90 triangle rules
Knowing your 30-60-90 triangle rules can save you time and energy on a variety of different math problems, especially a variety of geometry and trigonometry problems.
Geometry
Proper understanding of 30-60-90 triangles allows you to solve geometry questions that would be either impossible to solve or at least take a lot of time without knowing your ratio rules.
With special triangle ratios, you can find missing triangle heights or leg lengths (without having to use the Pythagorean Theorem), find the area of a triangle using missing height or base length information, and/or quickly calculate perimeters.
Anytime you need speed to answer a question, it comes in handy to remember shortcuts like your 30-60-90 rules.
Trigonometry
Memorizing and understanding your 30-60-90 triangles will also allow you to solve many trigonometric problems without needing a calculator or having to approximate your answers in decimal form.
A 30-60-90 triangle has fairly simple sines, cosines, and tangents for each angle (and these measurements will always be consistent).
The sine of 30° is always $1/2$.
The cosine of 60° is always $1/2$.
Although the other sine, cosine, and tangent are fairly straightforward, these two are the easiest to remember and are likely to be tested when it comes to standardized testing. So if you know these rules, you can find these trigonometric measurements as quickly as possible.
Remember the 30-60-90 rules
You know these 30-60-90 ratio rules are useful, but how do you keep the information in your head? Remembering the 30-60-90 triangle rules means remembering the 1:√3:2 ratio and knowing that the shortest side always corresponds to the shortest angle (30°) and the longest side always to the largest opposite angle (90°).
Some people memorize the ratio by saying, "x, 2x, x √3' because the sequence '1, 2, 3' is usually easy to remember. The only precaution when using this technique is to remember that the longest side is actually the one2x,notDieXmal√3.
Another way to remember your ratios is to use a mnemonic pun for the ratio 1:root 3:2 in order. For example: "Jackie Mitchell beat Lou Gehrig and 'won Ruthy too'" (one root three two). [Note: And it's a true fact of baseball history!]
Play around with your own mnemonics if they don't appeal to you - sing the relationship to a song, come up with your own first-root-three-two phrases, or come up with a relationship poem. You can even remember that a 30-60-90 triangle is half an equilateral triangle and figure out the measurements from there if you're not keen on memorizing them.
However, it is useful to remember these 30-60-90 rules and keep these ratios in mind for your future geometry and trigonometry questions.
Memorization is your friend, but you can make it happen.
Example 30-60-90 questions
Now that we've looked at the hows and whys of 30-60-90 triangles, let's look at some practice problems.
Geometry
A construction worker leans a 40-foot ladder against the side of a building at a 30-degree angle from the ground. The ground is level and the side of the building is perpendicular to the ground. How far does the ladder reach up the building to the next foot?
Without knowing our specific 30-60-90 triangle rules, we would have to use trigonometry and a calculator to find the solution to this problem since we only have one side measurement. But because we know this is a special triangle, we can find the answer in seconds.
If the building and the ground are perpendicular to each other, the building and the ground must form a right angle (90°). It also goes without saying that the ladder hits the ground at an angle of 30°. So we can see that the remaining angle must be 60°, making this a 30-60-90 triangle.
Now we know that the hypotenuse (longest side) of this 30-60-90 is 40 feet long, which means the shortest side is half that length. (Remember that the longest side is always twice the length of the shortest side.) Since the shortest side is opposite the 30° angle, and that angle is the degree of the ladder from the ground, this means the top of the ladder will hit the building 20 feet off the ground.
Our final answer is 20 feet.
Trigonometry
If in a right triangle sinΘ = $1/2$ and the shortest leg length is 8. What is the length of the missing side that is NOT the hypotenuse?
Now that you know your 30-60-90 rules, you can solve this problem without needing the Pythagorean theorem or a calculator.
We've been told that this is a right triangle, and we know from our special right triangle rules that sine is 30° = $1/2$. So the missing angle must be 60 degrees, making a 30-60-90 triangle.
And because this is a 30-60-90 triangle and we've been told that the shortest side is 8, the hypotenuse has to be 16 and the missing side has to be 8 * √3 or 8√3.
Our final answer is 8√3.
Die Take-Aways
Remembering the rules for 30-60-90 triangles will help you shorten a variety of math problems and free up time and energy for other types of questions. But remember that although knowing these rules is a handy tool, you can still solve your problems without them.
So keep the rules of in mindx, x√3, 2xand 30-60-90, whatever makes sense to you, and try to keep them straight whenever possible, but don't panic if your mind goes blank when the time comes. Either way, you have.
And if you need more practice, check this out30-60-90 triangle quiz. Have fun testing!