1. What Are Special Right Triangles?

Special right triangles are right-angled triangles with fixed angle measures and consistent side length ratios. The two most common types are the 45-45-90 and 30-60-90 triangles.

2. How Does the Calculator Work?

The calculator uses the fixed ratios of special right triangles:

Where:

• \( a \) — The basic side length unit
• \( \sqrt{2} \) ≈ 1.4142
• \( \sqrt{3} \) ≈ 1.7321

3. Properties of Special Right Triangles

45-45-90 Triangle: Both non-right angles are 45°. The legs are equal in length, and the hypotenuse is \( \sqrt{2} \) times the length of either leg.

30-60-90 Triangle: Angles are 30°, 60°, and 90°. The side opposite 30° is the shortest, the side opposite 60° is \( \sqrt{3} \) times the shortest side, and the hypotenuse is twice the shortest side.

4. Using the Calculator

Steps: Select the triangle type, enter the known side length, specify which side you entered (leg, hypotenuse, or long leg for 30-60-90), and click calculate.

5. Frequently Asked Questions (FAQ)

Q1: Why are these triangles special?
A: They have consistent angle measures and side ratios that make calculations predictable without trigonometry.

Q2: When are these triangles used?
A: In geometry, trigonometry, architecture, engineering, and any field requiring right angle calculations.

Q3: How accurate are the radical calculations?
A: The calculator provides exact radical forms where possible, with decimal approximations rounded to 2 decimal places.

Q4: Can I use this for non-right triangles?
A: No, this calculator is specifically for 45-45-90 and 30-60-90 right triangles.

Q5: What if I know an angle but not side lengths?
A: You need at least one side length to use these ratios. Angle measures alone aren't sufficient.