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.

Properties: The sides are always in the ratio 1 : √3 : 2 (shortest : medium : hypotenuse).

2. The 30-60-90 Triangle Formula

The side ratios are expressed as:

Where:

• \( x \) — Length of the shortest side (opposite 30° angle)
• \( x\sqrt{3} \) — Length of the medium side (opposite 60° angle)
• \( 2x \) — Length of the hypotenuse (opposite 90° angle)

Key Feature: Knowing just one side length allows you to calculate the other two sides.

3. Applications of 30-60-90 Triangles

Geometry: Used in trigonometry, engineering, and construction for calculating distances and angles.

Real-world: Common in roof trusses, staircase designs, and bridge construction.

4. Using the Calculator

Instructions: Enter the length of any known side (x value) in meters to calculate all three sides of the triangle.

5. Frequently Asked Questions (FAQ)

Q1: What if I know the hypotenuse instead of the short side?
A: Divide the hypotenuse by 2 to find x, then calculate the other sides.

Q2: How do I find the area of a 30-60-90 triangle?
A: Area = (1/2) × short side × medium side = (x × x√3)/2 = (x²√3)/2

Q3: Can I use this for any triangle?
A: No, this ratio only applies to triangles with 30-60-90 degree angles.

Q4: What are the angle ratios?
A: The angles are always in the ratio 1:2:3 (30°:60°:90°).

Q5: How accurate are these calculations?
A: The calculations are mathematically exact, limited only by measurement precision.