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

Definition: A special right triangle with angles measuring 30°, 60°, and 90° with consistent side length ratios.

Purpose: This triangle appears frequently in geometry and trigonometry problems, making its properties valuable to know.

2. How Does the Calculator Work?

The calculator uses the standard ratio formula:

Where:

• \( x \) — Shortest side (opposite 30° angle)
• \( x\sqrt{3} \) — Medium side (opposite 60° angle)
• \( 2x \) — Hypotenuse (opposite 90° angle)

Explanation: Given any one side length, the other two can be calculated using these fixed ratios.

3. Importance of 30-60-90 Triangles

Details: These triangles are fundamental in geometry, used in construction, engineering, and various mathematical proofs.

4. Using the Calculator

Tips: Enter the length of the shortest side (x) to calculate the other two sides. All values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: Why are the sides in this ratio?
A: The ratios come from the trigonometric functions of 30° and 60° angles in a right triangle.

Q2: Can I start with a different side?
A: Yes, but you'll need to adjust the formula accordingly (e.g., if given hypotenuse, divide by 2 to find x).

Q3: What units should I use?
A: Any consistent unit (meters, feet, inches) as the calculator preserves the ratio regardless of unit.

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

Q5: Where are these triangles used in real life?
A: In roof trusses, staircase designs, and any structure requiring 30° or 60° angles.