The Complete Guide to the 30-60-90 Triangle (2023)

The 30-60-90 triangle is a special right triangle, and knowing it can save you a lot of time on standardized tests like the SAT and ACT. Because its angles and aspect ratios are consistent, testers like to incorporate this triangle into problems, particularly in the non-calculator portion of the SAT. Here's what you need to know about the 30-60-90 triangle.

What is a 30-60-90 triangle?

A 30-60-90 triangle is a right triangle with angle measurements equal to 30º, 60º, and 90º(the right angle). Since the angles are always in this relationship, the sides are always in the same relationship to each other.

• The side opposite the 30º angle is the shortest and the length is usually denoted \(x\).
• The side opposite the 60º angle has a length equal to \(x\sqrt3\)
• The side opposite the 90º angle has the longest length and is equal to \(2x\)

Here's an example of a simple 30-60-90 triangle:

Knowing this ratio, you can easily identify missing information about a triangle without having to do more complicated calculations. With standardized tests, this can save you time in solving problems. When you see the relationship between angles and sides, you don't need to use triangle properties like the Pythagorean theorem.

Why 30-60-90 works

How do we know that the side lengths of the 30-60-90 triangle are always in the ratio \(1:\sqrt3:2\)? Although we can use a geometric proof, it's probably more helpful to check the properties of triangles, since knowing these properties will help you with other geometry and trigonometry problems.

Here are a few triangle properties to keep in mind:

1. In any triangle, the angles add up to 180º. In other words, if you know the measure of two of the angles, you can find the measure of the third by subtracting the measure of the two angles from 180.
2. In any triangle, the side opposite the smallest angle is always the shortest, while the side opposite the largest angle is always the longest. You can see how that applies to the 30-60-90 triangle above.
3. are triangles with equal degreessimilarand their sides are in equal proportion to each other. This means that all 30-60-90 triangles are similar, and we can use this information to solve similarity problems.

In addition, here are some triangle properties specific to right triangles:

• In right triangles, the side opposite 90ºAngle is called the hypotenuse, and the other two sides are the legs.
• In right triangles, the Pythagorean theorem explains the relationship between the legs and the hypotenuse: the sum of the length of each leg squared is equal to the length of the hypotenuse squared, or \(a^2+b^2=c^2\).

Based on this information if a problem says we have a right triangle and we are told that one of the angles is 30º, we can use the first property listed to know that the other angle is 60º. That's how 30-60-90 triangles often come up on standardized tests -- as a right triangle with an angle measure of 30º or 60º, and you have to figure out that it's 30-60-90.

Not only that, the right angle of a right triangle is always the largest angle - using Property 1 again, the other two angles must add up to 90º, which means that either of them cannot be greater than 90º. Since the right angle is always the largest angle, the property 2 hypotenuse is always the longest side.

We can use the Pythagorean theorem to show that the aspect ratio works with the 30-60-90 base triangle above.

\(a^2+b^2=c^2\)

\(1^2+(\sqrt3)^2=1+3=4=c^2\)

\(\sqrt4=2=c\)

With property 3 we know that all 30-60-90 triangles are similar and their sides are in the same ratio.

When to use 30-60-90 triangles?

For geometry problems:By knowing three pieces of information, one of which is that the triangle is a right triangle, we can easily solve missing pieces of information like angular measures and side lengths. With three pieces of information, usually two squares and 1 side, or 1 square and 2 sides, you can completely fill in the rest of the triangle.

For trigonometric problems:Once you know the basic definitions of sine, cosine, and tangent, it's very easy to find the values ​​for these from any 30-60-90 triangle. Sine, cosine, and tangent all represent a ratio of the sides of a triangle based on one of the angles denoted theta or \(\theta\). Because thatRelationshipof sides is the same for any 30-60-90 triangle, the sine, cosine, and tangent values ​​are always the same, particularly the following two, which are commonly used in standardized tests:

• Sine \(30\) is \(\frac{1}{2}\)
• Cosine \(60\) is \(\frac{1}{2}\)

Example 30-60-90 problems

To see the 30-60-90 in action, we've included a few problems that can be quickly solved with this particular right triangle.

sample geometry problem

When an angle of a right triangle is 30ºand the measure of the shortest side is 7, what is the measure of the remaining two sides?

• Answer: \(7\sqrt3\) and \(14\)

How to solve:Even though we may appear to be given only one angle measure, we are actually given two. Because it's a right triangle, we know that one of the angles is a right angle, or 90º, which means the other must be offset by 60º. This is a 30-60-90 triangle, and the sides have a ratio of \(x:x\sqrt3:2x\), where \(x\) is the length of the shortest side, in this case \(7\ ). The other sides must be \(7\:\cdot\:\sqrt3\) and \(7\:\cdot\:2\) or \(7\sqrt3\) and \(14\).

Example trigonometry problem

The triangle ABC has angular dimensions of 90, 30 and x. What is cosx?

• Answer: \(\frac{1}{2}\)

How to solve:We're given two angle measures so we can easily figure out that this is a 30-60-90 triangle. To find the cosine of an angle, we usually need the side lengths to find the ratio of the adjacent limb to the hypotenuse, but we know the ratio of the side lengths for any 30-60-90 triangle. The adjacent leg is always the shortest length, or \(1\), and the hypotenuse is always twice as long, for a ratio of \(1\) to \(2\), or \(\frac{1 {2 }\).

Example SAT problem

How long is AD in triangle ABC above?

• \(4\)

• \(6\)

• \(6\sqrt2\)

• \(6\sqrt3\)

Answer: \(6\)

How to solve:From the diagram we know that we are looking at two 30-60-90 triangles. The triangle BDC has two marked squares, 90º and 60º, so the third must be 30º. When line BD intersects line AC at 90º, then the lines are perpendicular, making Triangle BDA another 30-60-90 triangle.

Overall, triangle ABC is an equilateral triangle. We know this because the angles at A, B, and C are all 60º. From here we can use the knowledge that if AB is the hypotenuse and has length equal to \(12\), then AD is the shortest side and has half the length of the hypotenuse, or \(6\).

packaging

On the new SAT, you actually get the 30-60-90 triangle on the reference sheet at the beginning of each math section. While it's better to remember this triangle, you can always fall back on the sheet if needed, which can be reassuring when the pressure is on.

The best way to memorize the 30-60-90 triangle is to practice it when you have problems. Even if you use common practice problems, the more you use this triangle and the more variations of it you see, the more likely you will be able to identify it quickly on the SAT or ACT. Your math teacher may have some resources for practicing with 30-60-90.

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For more information on standardized tests and math tips, check out some of our other posts:

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