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## High school geometry

Course: high school geometry > unit 5.

- Special right triangles proof (part 1)
- Special right triangles proof (part 2)

## 30-60-90 triangle example problem

- Area of a regular hexagon
- Special right triangles review

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## Video transcript

## Choose Your Test

Sat / act prep online guides and tips, the easy guide to the 30-60-90 triangle.

General Education

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:

(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 $2√3$)

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.

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 $6√3$.

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.

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.

## 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.

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?

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.

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?

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!

Need more help with this topic? Check out Tutorbase!

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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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## 30-60-90 Triangle

A 30-60-90 triangle is a special right-angled triangle as the angles of the triangle are in the ratio 1:2:3. There are different types of triangles such as obtuse, isosceles, acute, equilateral, and so on. But only a few types of triangles are considered special triangles. These triangles are special as their sides and angles are consistent and predictable. Their properties can be used to solve various geometry or trigonometry problems. A 30-60-90 triangle—pronounced "thirty sixty ninety"—is one such very special type of triangle indeed.

In this lesson, we will explore the concept of the 30-60-90 triangle and learn all about it including its formula, definition, sides, area, and the rules that apply to this triangle.

## What is 30-60-90 Triangle?

The 30-60-90 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3. Here, a right triangle means being any triangle that contains a 90° angle. A 30-60-90 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°. Here are some of the variants of a 30-60-90 triangle. The triangles ABC and PQK are 30-60-90 triangles.

Here, in the triangle ABC, ∠ C = 30°, ∠ A = 60°, and ∠ B = 90° and in the triangle PQK, ∠ P = 30°, ∠ K = 60°, and ∠ Q = 90°

## 30-60-90 Triangle Sides

A 30-60-90 triangle is a special triangle since the length of its sides is always in a consistent relationship with one another.

In the below-given 30-60-90 triangle ABC, ∠ C = 30°, ∠ A = 60°, and ∠ B = 90°. We can understand the relationship between each of the sides from the below definitions:

- The side that is opposite to the 30° angle, AB = y will always be the smallest since 30° is the smallest angle in this triangle
- The side that is opposite to the 60° angle, BC = y × √ 3 = y √ 3 will be the medium length because 60° is the mid-sized degree angle in this triangle
- On the side that is opposite to the 90° angle, the hypotenuse AC = 2y will be the largest side because 90° is the largest angle.

The sides of a 30-60-90 triangle are always in the ratio of 1:√3: 2. This is also known as the 30-60-90 triangle formula for sides y: y√3: 2y. Let us learn the derivation of this ratio in the 30-60-90 triangle proof section. This formula can be verified using the Pythagoras theorem.

Consider some of the examples of a 30-60-90 degree triangle with these side lengths:

## 30-60-90-Triangle Theorem

The statement of the 30-60-90-Triangle Theorem is given as,

Statement: The length of the hypotenuse is twice the length of the shortest side and the length of the other side is √3 times the length of the shortest side in a 30-60-90-Triangle.

## 30-60-90-Triangle Formula

The above theorem can be written mathematically as the 30-60-90-Triangle Formula as 1:√3: 2 which is the ratio of the three sides of the 30-60-90-Triangle. Another formula for this special triangle 1:2:3 which is the ratio of the three angles of the 30-60-90-Triangle.

## 30-60-90-Triangle Proof

Let's consider an equilateral triangle ABC with a side length equal to 'a'.

Now, draw a perpendicular from vertex A to side BC at point D of the triangle ABC. The perpendicular in an equilateral triangle bisects the other side.

Triangle ABD and ADC are two 30-60-90 triangles. Both the triangles are similar and right-angled triangles. Hence, we can apply the Pythagoras theorem to find the length AD.

(AB) 2 = (AD) 2 + (BD) 2

a 2 = (AD) 2 + (a/2) 2

a 2 - (a/2) 2 = (AD) 2

3a 2 /4 = (AD) 2

(a √ 3)/2 = AD

AD = (a √ 3)/2

These sides also follow the same ratio a/2 : (a√3)/2: a

Multiply by 2 and divide by 'a',

(2a)/(2a) : (2a√3)/(2a): (2a/a)

We get 1:√3:2. This is the 30-60-90 triangle theorem.

## 30-60-90 Triangle Rule

In a 30-60-90 triangle, the measure of any of the three sides can be found out by knowing the measure of at least one side in the triangle. This is called the 30-60-90 triangle rule. The below-given table shows how to find the sides of a 30-60-90 triangle using the 30-60-90 triangle rule:

## Area of a 30-60-90 Triangle

The formula to calculate the area of a triangle is = (1/2) × base × height. In a right-angled triangle, the height is the perpendicular of the triangle. Thus, the formula to calculate the area of a right-angle triangle is = (1/2) × base × perpendicular

Let's learn how to apply this formula to find the area of the 30-60-90 triangle.

Base BC of the triangle is assumed to be 'a', and the hypotenuse of the triangle ABC is AC. We have learned in the previous section how to find the hypotenuse when the base is given.

Let's apply the formula we have learned.

Thus, perpendicular of the triangle = a/√3

Area of the triangle = (1/2) × base × perpendicular

Area = 1/2 × a × a/√3

Therefore, the area of the 30-60-90 triangle when the base (side of middle length) is given as 'a' is: a 2 /(2√3)

Related Articles

- Isosceles Triangles
- Obtuse Triangles
- Acute Triangle
- Perimeter of a Triangle

Important Notes on 30-60-90 Triangle

Here is a list of a few points that should be remembered while studying 30-60-90 triangles:

- The 30-60-90 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3 and the sides are in the ratio 1:√3: 2
- A 30-60-90 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°
- All the sides of a 30-60-90 triangle can be calculated if any one side is given. This is called the 30-60-90 triangle rule.

## Solved Examples on 30-60-90 Triangle

Example 1: Find the length of the hypotenuse of a right-angle triangle if the other two sides are 8 and 8 √ 3 units.

First, let's check the ratio to verify if it is suitable for a 30-60-90 triangle.

The ratio of the two sides = 8:8√3 = 1:√3

This indicates that the triangle is a 30-60-90 triangle. We know that the hypotenuse is 2 times the smallest side.

Thus, the hypotenuse is 2 × 8 = 16 units

Answer: Hypotenuse = 16 units

Example 2: A triangle has sides 2√2, 2√6, and 2√8. Find the angles of this triangle.

The sides of the triangle are 2√2, 2√6, and 2√8.

First, let's check whether the sides are following the 30-60-90 triangle rule.

2√2: 2√6: 2√8 can be re-written as 2√2: 2√2 × √3: 2 × 2√2

If we divide the ratio by 2√2, we get 1:√3: 2

These sides are following the 30-60-90 triangle rule. We know that the angles of a 30-60-90 triangle are 30°,60°, and 90°.

Answer: The angles of the triangle are 30°,60°, and 90°.

Example 3: Verify if the triangle is a 30-60-90 triangle if the sides of the triangle are 4 units, √48 units, and 8 units.

Solution: To verify if the triangle is a 30-60-90 triangle, we will check the ratio of the sides. We have

4 : √48 units : 8 = 4 : 4√3 : 4 × 2

= 1 : √3 : 2

Using the 30-60-90 triangle theorem, the sides are in the ratio 1 : √3 : 2. So the triangle is a 30-60-90 triangle.

Answer: The given triangle is a 30-60-90 triangle.

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## Practice Questions on 30-60-90 Triangle

Faqs on 30-60-90 triangle, what is a 30-60-90 triangle.

The 30-60-90 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3. A 30-60-90 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°.

## What Is the Perimeter of a 30-60-90 Triangle?

The perimeter of a 30 60 90 triangle with the smallest side equal to a is the sum of all three sides. The other two sides are a√3 and 2a. The perimeter of the triangle is a+a√3+2a = 3a+a√3 = a√3(1+√3).

## Are There Any Tips for Remembering the 30-60-90 Triangle Rules?

This method can be used to remember the 30-60-90 triangle rule. One can remember it as 1, 3, 2; it can resemble the ratio of the sides, all one needs to remember is that the middle term is √3

## What Are the Side Lengths of a 30-60-90 Triangle?

The sides of a 30-60-90 triangle have a set pattern. The side that is opposite to the 30° angle, 'y' will always be the smallest since 30° is the smallest angle in this triangle. The side that is opposite to the 60° angle, y√3 will be the medium length because 60° is the mid-sized degree angle in this triangle. The side that is opposite to the 90° angle, 2y will be the largest side because 90° is the largest angle.

## What Are the Rules for a 45-45-90 Triangle?

A 45-45-90 triangle has a right angle and two 45 degree angles. The two sides of a 45-45-90 triangle are always equal and the hypotenuse of the triangle is always opposite to the right angle.

## What Are Some Similarities Between 30-60-90 Triangles and 45-45-90 Triangles?

These are some of the similarities between 30-60-90 triangle and 45-45-90 triangle: both are not acute triangles, both are right-angle triangles, both are not obtuse triangles, the square of the hypotenuse equals the sum of the squares of the other two sides for both triangles, and the sum of the interior angles of both are 180°.

## Which Leg is the Long Leg in the 30-60-90 Triangle?

The long leg of a 30-60-90 Triangle is the leg whose length is greater than the shortest leg and ess than the hypotenuse. The length of the long leg is equal to √3 times the length of the shortest leg.

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## 30°-60°-90° Triangles

There is a special relationship among the measures of the sides of a 30 ° − 60 ° − 90 ° triangle.

In a 30 ° − 60 ° − 90 ° triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg.

To see why this is so, note that by the Converse of the Pythagorean Theorem , these values make the triangle a right triangle.

x 2 + ( x 3 ) 2 = x 2 + 3 x 2 = 4 x 2 = ( 2 x ) 2

## Download our free learning tools apps and test prep books

## 30-60-90 Right Triangles

In these lessons, we will learn

- the special right triangle called the 30-60-90 triangle.
- how to solve problems involving the 30-60-90 right triangle
- how to prove that the ratios between the sides of a 30-60-90 triangle are 1:√3:2.

Related Pages 45-45-90 Right Triangle Other Special Right Triangles More Geometry Lessons

Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right triangle is a right triangle whose sides are in a particular ratio . You can also use the Pythagorean theorem , but if you can see that it is a special triangle it can save you some calculations.

Here, we will look at the 30-60-90 triangle. We also have lessons on other special right triangles

## 30-60-90 Triangles

The 30-60-90 triangle is one example of a special right triangle. It is right triangle whose angles are 30°, 60° and 90°. The lengths of the sides of a 30-60-90 triangle are in the ratio of 1:√3:2.

The following diagram shows a 30-60-90 triangle and the ratio of the sides. Scroll down the page for more examples and solutions on how to use the 30-60-90 triangle.

The hypotenuse is always twice the length of the shorter leg (the side facing the 30° angle). The longer leg (the side facing the 60° angle) is √3 times of the shorter leg.

## Solve problems involving 30-60-90 right triangles

Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and 4√3 inches.

Solution: Step 1: Test the ratio of the lengths to see if it fits the n:n√3:2n ratio

4:4√3:? = n:n√3:2n

Step 2: Yes, it is a 30-60-90 triangle with n = 4

Step 3: Calculate the third side.

2 n = 2 × 4 = 8

Answer: The length of the hypotenuse is 8 inches.

You can also recognize a 30-60-90 triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30-60-90 special right triangle.

A right triangle with a 30°-angle or 60°-angle must be a 30-60-90 special right triangle.

Example 2: Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 8 inches and one of the angles is 30°.

Solution: This is a right triangle with a 30-60-90 triangle.

You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio n:n√3:2n, we get that 2 n = 8 ⇒ n = 4.

Substituting n = 4 into the first and second value of the ratio we get that the other two sides are 4 and 4√3.

Answer: The lengths of the two sides are 4 inches and 4√3 inches.

How to solve a 30-60-90 triangle given the length of one side?

Special Right Triangles in Geometry: 45-45-90 and 30-60-90 degree triangles This video discusses two special right triangles, how to derive the formulas to find the lengths of the sides of the triangles by knowing the length of one side, and then does a few examples using them.

Applying the 30-60-90 triangle to find the height of a building An example in which we use some of the great properties of a 30-60-90 right triangle to find the height of a tower.

Using what we know about 30-60-90 triangles to solve what at first seems to be a challenging problem

## Proof the ratios between the sides of a 30-60-90 triangle

Proving the ratios between the sides of a 30-60-90 triangle are 1:√3:2.

Prove how the side lengths of a 30-60-90 triangle are related and then use that relationship to quickly find side lengths

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

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- 30°-60°-90° Triangle – Explanation & Examples

## 30°-60°-90° Triangle – Explanation & Examples

It also carries equal importance to the 45°-45°-90° triangle due to the relationship of its side. It has two acute angles and one right angle.

## What is a 30-60-90 Triangle?

A 30-60-90 triangle is a special right triangle whose angles are 30º, 60º, and 90º. The triangle is special because its side lengths are always in the ratio of 1: √3:2.

Any triangle of the form 30-60-90 can be solved without applying long-step methods such as the Pythagorean Theorem and trigonometric functions.

The easiest way to remember the ratio 1: √3: 2 is to memorize the numbers; “1, 2, 3” . One precaution to using this mnemonic is to remember that 3 is under the square root sign.

From the illustration above, we can make the following observations about the 30-60-90 triangle:

- The shorter leg, which is opposite to the 30- degree angle, is labeled as x.
- The hypotenuse, which is opposite to the 90-degree angle, is twice the shorter leg length (2x).
- The longer leg, which is opposite to the 60-degree angle, is equal to the shorter leg’s product and the square root of three (x√3).

## How to Solve a 30-60-90 Triangle?

Solving problems involving the 30-60-90 triangles, you always know one side, from which you can determine the other sides. For that, you can multiply or divide that side by an appropriate factor.

You can summarize the different scenarios as:

- When the shorter side is known, you can find the longer side by multiplying the shorter side by a square root of 3. After that, you can apply Pythagorean Theorem to find the hypotenuse.
- When the longer side is known, you can find the shorter side by diving the longer side by the square root of 3. After that, you can apply Pythagorean Theorem to find the hypotenuse.
- When the shorter side is known, you can find the hypotenuse by multiplying the shorter side by 2. After that, you can apply Pythagorean Theorem to find the longer side.
- When the hypotenuse is known, you can find the shorter side by dividing the hypotenuse by 2. After that, you can apply Pythagorean Theorem to find the longer side.

This means the shorter side acts as a gateway between the other two sides of a right triangle . You can find the longer side when the hypotenuse is given or vice versa, but you always have to find the shorter side first.

Also, to solve the problems involving the 30-60-90 triangles , you need to be aware of the following properties of triangles:

- The sum of interior angles in any triangle adds up to 180º. Therefore, if you know the measure of two angles, you can easily determine the third angle by subtracting the two angles from 180 degrees.
- The shortest and longest sides in any triangle are always opposite to the smallest and largest angles. This rule also applies to the 30-60-90 triangle.
- Triangles with the same angle measures are similar, and their sides will always be in the same ratio to each other. The concept of similarity can therefore be used to solve problems involving the 30-60-90 triangles.
- Since the 30-60-90 triangle is a right triangle, then the Pythagorean theorem a 2 + b 2 = c 2 is also applicable to the triangle. For instance, we can prove the hypotenuse of the triangle is 2x as follows:

⇒ c 2 = x 2 + (x√3) 2

⇒ c 2 = x 2 + (x√3) (x√3)

⇒ c 2 = x 2 + 3x 2

⇒ c 2 = 4x 2

Find the square root of both sides.

√c 2 = √4x 2

Hence, proved.

Let’s work through some practice problems.

A right triangle whose one angle is 60 degrees has the longer side as 8√3 cm. Calculate the length of its shorter side and the hypotenuse.

From the ratio x: x√3: 2x, the longer side is x√3. So, we have;

x√3 = 8√3 cm

Square both sides of the equation.

⇒ (x√3) 2 = (8√3) 2

⇒ 3x 2 = 64 * 3

Find the square of both sides.

Substitute.

2x = 2 * 8 = 16 cm.

Hence, the shorter side is 8 cm, and the hypotenuse is 16 cm.

A ladder leaning against a wall makes an angle of 30 degrees with the ground. If the length of the ladder is 9 m, find;

a. The height of the wall.

b. Calculate the length between the foot of the ladder and the wall.

One angle is 30 degrees; then this must be a 60°- 60°- 90°right triangle.

Ratio = x: x√3: 2x.

a. The height of the wall = 4.5 m

b. x√3 = 4.5√3 m

The diagonal of a right triangle is 8 cm. Find the lengths of the other two sides of the triangle given that one of its angles is 30 degrees.

This is must be a 30°-60°-90° triangle. Therefore, we use the ratio of x: x√3:2x.

Diagonal = hypotenuse = 8cm.

x√3 = 4√3 cm

The shorter side of the right triangle is 4cm, and the longer side is 4√3 cm.

Find the value of x and z in the diagram below:

The length measuring 8 inches will be the shorter leg because it is opposite the 30-degree angle. To find the value of z (hypotenuse) and y (longer leg), we proceed as follows;

From the ratio x: x√3:2x;

x = 8 inches.

⇒ x√3 = 8√3

⇒2x = 2(8) = 16.

Hence, y = 8√3 inches and z = 16 inches.

If one angle of a right triangle is 30º and the shortest side’s measure is 7 m, what is the measure of the remaining two sides?

This is a 30-60-90 triangle in which the side lengths are in the ratio of x: x√3:2x.

Substitute x = 7m for the longer leg and the hypotenuse.

⇒ x √3 = 7√3

⇒ 2x = 2(7) =14

Hence, the other sides are 14m and 7√3m

In a right triangle, the hypotenuse is 12 cm, and the smaller angle is 30 degrees. Find the length of the long and short leg.

Given the ratio of the sides = x: x√3:2x.

Substitute x = 6 cm for the long and short leg to get;

Short leg = 6cm.

long leg = 6√3 cm

The two sides of a triangle are 5√3 mm and 5mm. Find the length of its diagonal.

Test the ratio of the side lengths if it fits the x: x√3:2x ratio.

5: 5√3:? = 1(5): √3 (5):?

Therefore, x = 5

Multiply 2 by 5.

2x = 2* 5 = 10

Hence, the hypotenuse is equal to 10 mm.

A ramp that makes an angle of 30 degrees with the ground is used to offload a lorry that is 2 feet high. Calculate the length of the ramp.

This must be a 30-60-90 triangle.

x = 2 feet.

2x = 4 feet

Hence, the length of the ramp is 4 feet.

Find the hypotenuse of a 30°- 60°- 90° triangle whose longer side is 6 inches.

Ratio = x: x√3:2x.

⇒ x√3 = 6 inches.

Square both sides

⇒ (x√3) 2 = 36

⇒ 3x 2 = 36

x = 2√3 inches.

Practice Problems

- In a 30°- 60°- 90° triangle, let the side across from the 60° angle is given as 9√3. Find the length of the other two sides.
- If the hypotenuse of the 30°- 60°- 90° triangle is 26, find the other two sides.
- If the longer side of a 30°- 60°- 90° triangle is 12, what is the sum of the other two sides of this triangle?

## Practice Questions

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