Side Lengths of 30-60-90 Triangle

30-60-90 Triangle Formula:

\[ x : x\sqrt{3} : 2x \]

(opposite 30° : opposite 60° : hypotenuse)

Shortest side (opposite 30°):

Medium side (opposite 60°):

Hypotenuse (opposite 90°):

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1. What is a 30-60-90 Triangle?

Definition: A special right triangle where the angles measure 30°, 60°, and 90° with side lengths in the ratio 1 : √3 : 2.

Properties: The sides opposite these angles follow the pattern x (shortest), x√3 (medium), and 2x (hypotenuse).

2. How Does the Calculator Work?

The calculator uses the formula:

\[ x : x\sqrt{3} : 2x \]

Where:

x — Length of the side opposite the 30° angle (shortest side)
x√3 — Length of the side opposite the 60° angle
2x — Length of the hypotenuse (opposite the 90° angle)

Explanation: The calculator determines x based on which side you know, then calculates the other two sides using the ratio.

3. Applications of 30-60-90 Triangles

Details: These triangles are commonly used in trigonometry, engineering, architecture, and various construction projects.

4. Using the Calculator

Tips: Enter the length of any one side and specify which side it is (shortest, medium, or hypotenuse). The calculator will determine the other two sides.

5. Frequently Asked Questions (FAQ)

Q1: Why is the ratio 1 : √3 : 2?
A: This ratio comes from the trigonometric relationships in a 30-60-90 triangle, derived from the unit circle and special angle values.

Q2: Can I use this for any triangle?
A: No, this calculator only works for 30-60-90 triangles. Other triangles have different side length relationships.

Q3: What units should I use?
A: Any consistent unit (meters, feet, inches, etc.). The results will be in the same units as your input.

Q4: How accurate are the results?
A: Results are mathematically precise, though displayed with 3 decimal places for readability.

Q5: What if I know the area instead of a side?
A: First calculate a side from the area (Area = (x²√3)/2), then use this calculator.