Special Right Triangles Calculator with Radicals

Special Right Triangles:

\[ \text{45-45-90: } 1 : 1 : \sqrt{2} \] \[ \text{30-60-90: } 1 : \sqrt{3} : 2 \]

Triangle Type:

Known Side Length:

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Which Side:

Triangle Side Lengths:

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1. What Are Special Right Triangles?

Special right triangles are right-angled triangles with angle measures and side lengths that follow consistent 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:

\[ \text{45-45-90 Triangle: } \text{Leg} : \text{Leg} : \text{Hypotenuse} = 1 : 1 : \sqrt{2} \] \[ \text{30-60-90 Triangle: } \text{Short Leg} : \text{Long Leg} : \text{Hypotenuse} = 1 : \sqrt{3} : 2 \]

Where:

45-45-90 — Two equal legs and angles, hypotenuse is leg × √2
30-60-90 — Short leg opposite 30°, long leg is short leg × √3, hypotenuse is 2 × short leg

Explanation: These ratios come from the trigonometric properties of these angle combinations.

3. Importance of Special Right Triangles

Details: These triangles are fundamental in geometry, trigonometry, and many real-world applications like construction and engineering. They simplify calculations involving right angles.

4. Using the Calculator

Tips: Select the triangle type, enter the known side length, and specify which side it represents (leg, hypotenuse, etc.). The calculator will compute all other sides.

5. Frequently Asked Questions (FAQ)

Q1: Why are these triangles special?
A: They have consistent side ratios that make calculations easier without needing trigonometry functions.

Q2: How do I remember the ratios?
A: For 45-45-90, both legs are equal. For 30-60-90, remember 1 (shortest), √3 (middle), 2 (longest).

Q3: Can I use this for non-right triangles?
A: No, these specific ratios only apply to right triangles with these exact angle measures.

Q4: Are the radical forms exact?
A: Yes, the calculator provides exact radical forms where possible, with decimal approximations for reference.

Q5: Where are these triangles used in real life?
A: In construction (roof pitches, stair angles), engineering, navigation, and anywhere right angles with specific proportions are needed.