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

Definition: A special right triangle with angles of 30°, 60°, and 90° whose sides are always in the ratio 1 : √3 : 2.

Purpose: This calculator helps determine all side lengths when one side is known, useful in geometry, construction, and design.

2. How Does the Calculator Work?

The calculator uses the fixed ratio:

Where:

• 1 — Shortest side (opposite 30° angle)
• √3 — Medium side (opposite 60° angle)
• 2 — Hypotenuse (opposite 90° angle)

Explanation: Based on which side you know, the calculator scales the other sides proportionally using these ratios.

3. Importance of 30-60-90 Triangles

Details: These triangles are fundamental in geometry, appear frequently in trigonometry problems, and are useful in construction for creating precise angles.

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 are the sides in this exact ratio?
A: This ratio comes from the properties of equilateral triangles cut in half, maintaining consistent trigonometric relationships.

Q2: How accurate are these ratios?
A: These are exact mathematical ratios, not approximations. √3 is approximately 1.732 but the calculator uses the exact value.

Q3: Can I use this for any triangle?
A: No, this only works for 30-60-90 triangles. Other right triangles have different side ratios.

Q4: What units should I use?
A: Any consistent unit (inches, feet, meters). The results will be in the same units as your input.

Q5: How is this useful in real-world applications?
A: Useful in construction for roof framing, stair stringers, and any project requiring precise 30° or 60° angles.