What is a 30 60 90 triangle

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Hypotenuse equals twice the smallest leg, while the larger leg is sqrt(3) times the smallest.

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The 30-60-90 triangle is shaped like half of an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30 degrees, 60 degrees, and 90 degrees, thus, its name! In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3.

The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle.

Two of the most common right triangles are 30-60-90 and the 45-45-90-degree triangles. All 30-60-90 triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in radians, it translates to the following:

The figure illustrates the ratio of the sides for the 30-60-90-degree triangle.

A 30-60-90-degree right triangle

If you know one side of a 30-60-90 triangle, you can find the other two by using shortcuts. Here are the three situations you come across when doing these calculations:

• Type 1: You know the short leg (the side across from the 30-degree angle). Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg.

• Type 2: You know the hypotenuse. Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg.

• Type 3: You know the long leg (the side across from the 60-degree angle). Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse.

  
  Finding the other sides of a 30-60-90 triangle when you know the hypotenuse

In the triangle TRI in this figure, the hypotenuse is 14 inches long; how long are the other sides?

Because you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side:

About This Article

This article is from the book:

• Pre-Calculus For Dummies ,

About the book author:

Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

This article can be found in the category:

• Pre-Calculus ,

What is a 30-60-90 Triangle?

A 30-60-90 triangle is a right triangle where the three interior angles measure 30°, 60°, and 90°.

Right triangles with 30-60-90 interior angles are known as special right triangles. Special triangles in geometry because of the powerful relationships that unfold when studying their angles and sides.

In all triangles, the relationships between angles and their opposite sides are easy to understand. The greater the angle, the longer the opposite side.

This means, of the three interior angles, the largest interior angle is opposite the longest of the three sides, and the smallest angle will be opposite the shortest side.

In a right triangle, recall that the side opposite the right angle (the largest angle) is called the hypotenuse (the longest side, and the other two sides are called legs.

Table Of Contents

1. What is a 30-60-90 Triangle?
2. 30-60-90 Triangle Theorem
3. How to Solve a 30-60-90 Triangle
   • 30-60-90 Triangle Rules
4. 30-60-90 Triangle Examples

30-60-90 Triangle Ratio

A 30-60-90 degree triangle is a special right triangle, so it's side lengths are always consistent with each other. The ratio of the sides follow the 30-60-90 triangle ratio:

• Short side (opposite the 30 degree angle) = x
• Hypotenuse (opposite the 90 degree angle) = 2x
• Long side (opposite the 60 degree angle) = x3

30-60-90 Triangle Theorem

These three special properties can be considered the 30-60-90 triangle theorem and are unique to these special right triangles:

1. The hypotenuse (the triangle's longest side) is always twice the length of the short leg
2. The length of the longer leg is the short leg's length times 3
3. If you know the length of any one side of a 30-60-90 triangle, you can find the missing side lengths

Other interesting properties of 30-60-90 triangles are:

• All 30-60-90 triangles are similar
• Two 30-60-90 triangles sharing a long leg form an equilateral triangle

How to Solve a 30-60-90 Triangle

Education is knowing that 30-60-90 triangles have three properties laid out in the theorem. Wisdom is knowing what to do with that knowledge. Suppose you have a 30-60-90 triangle:

We know that the hypotenuse of this triangle is twice the length of the short leg:

3.46 km2 = 1.73 km

We also know that the long leg is the short leg multiplied times the square  root of 3:

1.73 × 3 ≈ 3 km

We set up our special 30-60-90 to showcase the simplicity of finding the length of the three sides. Try figuring this one out:

The long leg is the short leg times 3, so can you calculate the short leg's length? Did you say 5?

The length of the hypotenuse is always twice the short leg's length. Did you get 10?

You can create your own 30-60-90 Triangle formula using the known information in your problem and the following rules. This table of 30-60-90 triangle rules to help you find missing side lengths:

30-60-90 Triangle Rules

When working with 30-60-90 triangles, you may be tempted to force a relationship between the hypotenuse and the long leg. That relationship is challenging because of the square root of 3.

Work carefully, concentrating on the relationship between the hypotenuse and short leg, then short leg and long leg.

You will notice our examples so far only provided information that would "plugin" easily using our three properties. Sometimes the geometry is not so easy.

What if the long leg is labeled with a simple, whole number?

You leap into the problem since getting the short leg is simply a matter of dividing the long leg by the square root of 3, then doubling that to get the hypotenuse.

But you cannot leave the problem like this:

27 cm3

The rules of mathematics do not permit a radical in the denominator, so you must rationalize the fraction. Multiply both numerator and denominator times 3:

27 cm3 × 33

This simplifies to:

27 39 = 27  33 = 9 3

Unless your directions are to provide a decimal answer, this can be your final answer for the length of the short side. Doubling this gives 18 3 for the hypotenuse.

Another warning flag with 30-60-90 triangles is that you can become so engrossed in the three properties that you lose sight of the triangle itself. It is still a triangle, so its interior angles must add to 180°, and its three sides must still adhere to the Pythagorean Theorem:

You can use the Pythagorean Theorem to check your work or to jump-start a solution.

30-60-90 Triangle Examples

A right triangle has a short side with a length of 14 meters with the opposite angle measuring 30°. What are the other two lengths?

We know immediately that the triangle is a 30-60-90, since the two identified angles sum to 120 °:

180° - 120° = 60°

The missing angle measures 60°. It follows that the hypotenuse is 28 m, and the long leg is 14 m * 3.

What is you have a triangle with the hypotenuse labeled 2,020 mm, the short leg labeled 1,010 mm, and the long leg labeled 1,0103.

Your knowledge of the 30-60-90 triangle will help you recognize this immediately. You can confidently label the three interior angles because you see the relationships between the hypotenuse and short leg and the short leg and long leg.

Here is a more elaborate problem. What is the length of the shorter leg, line segment MH ?:

Did you say 50 inches? This is really two 30-60-90 triangles, which means hypotenuse MA is also 100 inches, which means the shortest leg MH is 50 inc hes.

Next Lesson:

Triangle Congruence Theorem

What is the formula for 30 60 90 triangle?