30-60-90 Special Right Triangles

30-60-90 Triangles

A 30-60-90 triangle is a special type of right triangle where the angles are 30°, 60°, and 90°. The sides of a 30-60-90 triangle follow a consistent ratio of 1:√3:2 making calculations quick and easy. If we are given one side of the triangle, we are able to easily find the other sides using the ratio of 1:√3:2.

Another special right triangle is the 45-45-90 triangle.

The following diagram shows the properties of a 30-60-90 Triangle. Scroll down the page for more examples and solutions on how to solve 30-60-90 Triangles.

Geometry Worksheets
Practice your skills with the following geometry worksheets:
Printable & Online Geometry Worksheets

Side Length Ratio: 1:√3:2
Shortest side (opposite 30°) = x
Middle side (opposite 60°) = x√3
Hypotenuse (opposite 90°) = 2x

How to Use 30-60-90 Triangles
If you know the length of just one side of a 30-60-90 triangle, you can find the lengths of the other two sides using these ratios.
Case 1: Given the shortest side (x)
Hypotenuse = 2x
Medium side = x√3
Example: If the side opposite the 30° angle is 7, then the hypotenuse is 2×7=14, and the side opposite the 60° angle is 7√3.

Case 2: Given the hypotenuse (2x)
Shortest side = x
Medium side = x√3
Example: If the hypotenuse is 10, then the side opposite the 30° angle = 10/2 = 5 and the side opposite the 60° angle is 5√3.

Case 3: Given the medium side (x√3)
Shortest side = x
Hypotenuse = 2x
Example: If the side opposite the 60° angle is 6√3, then the side opposite the 30° angle is 6. The hypotenuse is 2×6=12.

Importance

• Efficiency: It allows you to solve for unknown side lengths much faster than using the Pythagorean theorem or trigonometric functions (sine, cosine, tangent) for every problem.
• Exact Values: It provides exact radical answers, which are often preferred over decimal approximations.
• Trigonometry: The side ratios are the basis for exact trigonometric values for 30° and 60° (e.g., sin(30°)=1/2, cos(60°)=1/2, tan(30°)=1/3).
• Problem Solving: It’s frequently encountered in standardized tests and real-world problems involving geometry and right triangles.

How to find the legs and hypotenuse in 30-60-90 triangles when given: the short leg, the long leg, or the hypotenuse.

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Solve a 30-60-90 Right Triangle
This video provides examples of how to solve a 30-60-90 triangle given the length of one side.

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30-60-90 Triangle

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Finding the Missing Side of 30-60-90 Triangle

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