Special Right Triangles Calculator
Calculate all measurements for 45-45-90, 30-60-90, and other special right triangles
Calculate Special Right Triangle
Isosceles right triangle with equal legs
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Enter the known side length
45°-45°-90° Triangle
Side Ratio: 1 : 1 : √2
Equal legs, hypotenuse = leg × √2
Angles: 0.0°, 0.0°, 90°
Triangle Measurements
Sides
Angles
Properties
Example Calculation
45-45-90 Triangle Example
Given: One leg = 5 units
Find: Other leg, hypotenuse, area, perimeter
Solution:
• Other leg = 5 units (equal legs)
• Hypotenuse = 5√2 ≈ 7.07 units
• Area = (5 × 5)/2 = 12.5 square units
• Perimeter = 5 + 5 + 7.07 ≈ 17.07 units
Common Special Triangles
45-45-90
Isosceles right triangle
Ratio: 1 : 1 : √2
30-60-90
Half equilateral triangle
Ratio: 1 : √3 : 2
3-4-5
Pythagorean triple
Integer sides
Triangle Tips
Special triangles have exact ratios that make calculations easier
45-45-90 triangles appear when cutting squares diagonally
30-60-90 triangles are half of equilateral triangles
Pythagorean triples have integer side lengths
Understanding Special Right Triangles
What Makes Triangles "Special"?
Special right triangles have side lengths and angles that follow predictable patterns, making calculations easier without needing the Pythagorean theorem or trigonometric functions.
Types of Special Triangles
- •Angle-based: 45-45-90 and 30-60-90 triangles
- •Side-based: Pythagorean triples like 3-4-5
- •Geometric: Triangles from geometric constructions
Common Applications
- •Architecture and construction
- •Engineering design
- •Navigation and surveying
- •Computer graphics and game development
Memory Tip: Remember 45-45-90 as "1-1-√2" and 30-60-90 as "1-√3-2"