Is There an Easy Way to Determine the Shorter Side of a Triangle Given the Measure of the Angles

Acute, obtuse, isosceles, equilateral…When it comes to triangles, there are many different varieties, but only a choice few that are "special." These special triangles have sides and angles which 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"—happens to be a very special type of triangle indeed.

In this guide, we'll walk you through what a 30-60-90 triangle is, why it works, and when (and how) to use your knowledge of it. So let's get to it!

What Is a 30-60-90 Triangle?

A 30-60-90 triangle is a special right triangle (a right triangle being any triangle that contains a 90 degree angle) that always has degree angles of 30 degrees, 60 degrees, and 90 degrees. Because it is a special triangle, it also has side length values which are always in a consistent relationship with one another.

The basic 30-60-90 triangle ratio is:

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 could have side lengths of:

2, 2√3, 4

7, 7√3, 14

√3, 3, 2√3

(Why is the longer leg 3? In this triangle, the shortest leg ($x$) is $√3$, so for the longer leg, $x√3 = √3 * √3 = √9 = 3$. And the hypotenuse is 2 times the shortest leg, or $2√3$)

And so on.

The side opposite the 30° angle is always the smallest, because 30 degrees is the smallest angle. The side opposite the 60° angle will be the middle length, because 60 degrees is the mid-sized degree angle in this triangle. And, finally, the side opposite the 90° angle will always be the largest side (the hypotenuse) because 90 degrees is the largest angle.

Though it may look similar to other types of right triangles, the reason a 30-60-90 triangle is so special is that you only need three pieces of information in order to find every other measurement. So long as you know the value of two angle measures and one side length (doesn't matter which side), you know everything you need to know about your triangle.

For example, we can use the 30-60-90 triangle formula to fill in all the remaining information blanks of the triangles below.

Example 1

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 smaller given side is opposite the 30°.

The longer leg must, therefore, be opposite the 60° angle and measure $6 * √3$, or $6√3$.

Example 2

We can see that this must be a 30-60-90 triangle because we can see that this is a right triangle with one given measurement, 30°. The unmarked angle must then be 60°.

Since 18 is the measure opposite the 60° angle, it must be equal to $x√3$. The shortest leg must then measure $18/√3$.

(Note that the leg length will actually be $18/{√3} * {√3}/{√3} = {18√3}/3 = 6√3$ because a denominator cannot contain a radical/square root).

And the hypotenuse will be $2(18/√3)$

(Note that, again, you cannot have a radical in the denominator, so the final answer will really be 2 times the leg length of $6√3$ => $12√3$).

Example 3

Again, we are given two angle measurements (90° and 60°), so the third measure will be 30°. Because this is a 30-60-90 triangle and the hypotenuse is 30, the shortest leg will equal 15 and the longer leg will equal 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 special triangle work the way it does? How do we know these rules are legit? Let's walk through exactly how the 30-60-90 triangle theorem works and prove why these side lengths will always be consistent.

First, let's forget about 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. Because a triangle's interior angles always add up to 180° and $180/3 = 60$, an equilateral triangle will always have three 60° angles.

Now let's drop down a height from the topmost angle to the base of the triangle.

We've now created two right angles and two congruent (equal) triangles.

How do we know they're equal triangles? Because we dropped a height from an equilateral triangle, we've split the base exactly in half. The new triangles also share one side length (the height), and they each have the same hypotenuse length. Because they share three side lengths in common (SSS), this means the triangles are congruent.

Note: not only are the two triangles congruent based on the principles of side-side-side lengths, or SSS, but also based on side-angle-side measures (SAS), angle-angle-side (AAS), and angle-side-angle (ASA). Basically? They're most definitely congruent.

Now that we've proven the congruencies of the two new triangles, we can see that the top angles must each be equal to 30 degrees (because each triangle already has angles of 90° and 60° and must add up to 180°). This means we have made two 30-60-90 triangles.

And because we know that we cut the base of the equilateral triangle in half, 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 us call our original side length $x$ and our bisected length $x/2$.

Now all that leaves us to do is to find our mid-side length that the two triangles share. To do this, we can simply use the Pythagorean theorem.

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

$(x/2)^2 + b^2 = x^2$

$b^2 = x^2 - ({x^2}/4)$

$b^2 = {4x^2}/4 - {x^2}/4$

$b^2 = {3x^2}/4$

$b = {√3x}/2$

So we're left with: $x/2, {x√3}/2, x$

Now let's multiply each measure by 2, just to make life easier and avoid all the fractions. That way, we're left with:

$x$, $x√3$, $2x$

We can see, therefore, that a 30-60-90 triangle will always have consistent side lengths of $x$, $x√3$, and $2x$ (or $x/2$, ${√3x}/2$, and $x$).

Luckily for us, we can prove 30-60-90 triangle rules true without all of...this.

When to Use 30-60-90 Triangle Rules

Knowing the 30-60-90 triangle rules will be able to save you time and energy on a multitude of different math problems, namely a wide variety of geometry and trigonometry problems.

Geometry

Proper understanding of the 30-60-90 triangles will allow you to solve geometry questions that would either be impossible to solve without knowing these ratio rules, or at the very least, would take considerable time and effort to solve the "long way."

With the special triangle ratios, you can figure out missing triangle heights or leg lengths (without having to use the Pythagorean theorem), find the area of a triangle by using missing height or base length information, and quickly calculate perimeters.

Any time you need speed to answer a question, remembering shortcuts like your 30-60-90 rules will come in handy.

Trigonometry

Memorizing and understanding the 30-60-90 triangle ratio will also allow you to solve many trigonometry problems without either the need for a calculator or the need 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).

Sine of 30° will always be $1/2$.

Cosine of 60° will always be $1/2$.

Though the other sines, cosines, and tangents are fairly simple, these are the two that are the easiest to memorize and are likely to show up on tests. So knowing these rules will allow you to find these trigonometry measurements as quickly as possible.

Tips for Remembering 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 is a matter of remembering the ratio of 1: √3 : 2, and knowing that the shortest side length is always opposite the shortest angle (30°) and the longest side length is always opposite the largest angle (90°).

Some people memorize the ratio by thinking, "$\bi x$, $\bo 2 \bi x$, $\bi x \bo √ \bo3$," because the "1, 2, 3" succession is typically easy to remember. The one precaution to using this technique is to remember that the longest side is actually the $2x$, not the $x$ times $√3$.

Another way to remember your ratios is to use a mnemonic wordplay on the 1: root 3: 2 ratio in their proper order. For example, "Jackie Mitchell struck out Lou Gehrig and 'won Ruthy too,'": one, root three, two. (And it's a true baseball history fact to boot!)

Play around with your own mnemonic devices if these don't appeal to you—sing the ratio to a song, find your own "one, root three, two" phrases, or come up with a ratio poem. You can even just remember that a 30-60-90 triangle is half an equilateral and figure out the measurements from there if you don't like memorizing them.

However it makes sense to you to remember these 30-60-90 rules, keep those ratios your head for your future geometry and trigonometry questions.

Memorization is your friend, however you can make it happen.

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Example 30-60-90 Questions

Now that we've looked at the hows and whys of 30-60-90 triangles, let's work through some practice problems.

Geometry

A construction worker leans a 40-foot ladder up against the side of a building at an angle of 30 degrees off the ground. The ground is level and the side of the building is perpendicular to the ground. How far up the building does the ladder reach, to the nearest foot?

Without knowing our 30-60-90 special 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 of a triangle. But because we know that this is a special triangle, we can find the answer in just seconds.

If the building and the ground are perpendicular to one another, that must mean the building and the ground form a right (90°) angle. It's also a given that the ladder meets the ground at a 30° angle. We can therefore see that the remaining angle must be 60°, which makes this a 30-60-90 triangle.

Now we know that the hypotenuse (longest side) of this 30-60-90 is 40 feet, which means that the shortest side will be half that length. (Remember that the longest side is always twice—$2x$—as long as the shortest side.) Because the shortest side is opposite the 30° angle, and that angle is the degree measure of the ladder from the ground, that means that the top of the ladder hits the building 20 feet off the ground.

Our final answer is 20 feet.

Trigonometry

If, in a right triangle, sin Θ = $1/2$ and the shortest leg length is 8. What is the length of the missing side that is NOT the hypotenuse?

Because you know your 30-60-90 rules, you can solve this problem without the need for either the pythagorean theorem or a calculator.

We were told that this is a right triangle, and we know from our special right triangle rules that sine 30° = $1/2$. The missing angle must, therefore, be 60 degrees, which makes this a 30-60-90 triangle.

And because this is a 30-60-90 triangle, and we were told that the shortest side is 8, the hypotenuse must be 16 and the missing side must be $8 * √3$, or $8√3$.

Our final answer is 8√3.

The Take-Aways

Remembering the rules for 30-60-90 triangles will help you to shortcut your way through a variety of math problems. But do keep in mind that, while knowing these rules is a handy tool to keep in your belt, you can still solve most problems without them.

Keep track of the rules of $x$, $x√3$, $2x$ and 30-60-90 in whatever way makes sense to you and try to keep them straight if you can, but don't panic if your mind blanks out when it's crunch time. Either way, you've got this.

And, if you need more practice, go ahead and check out this 30-60-90 triangle quiz. Happy test-taking!

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About the Author

Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.

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Source: https://blog.prepscholar.com/30-60-90-triangle-ratio-formula

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