Pythagorean Theorem Calculator
Professional Pythagorean theorem calculator for right triangles. Calculate missing sides, verify right triangles, and find distances between points with step-by-step solutions.
Find Hypotenuse (c)
Find Missing Leg
Verify Right Triangle
Distance Between Points
Understanding the Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Pythagorean Formula
- Formula: a² + b² = c²
- Where: a and b are the legs, c is the hypotenuse
- Hypotenuse: The longest side, opposite the right angle
- Legs: The two shorter sides that form the right angle
Applications
| Application | Description | Example |
|---|---|---|
| Construction | Measuring distances and heights | Ladder against wall |
| Navigation | Finding shortest paths | GPS coordinates |
| Engineering | Structural calculations | Roof pitch, ramps |
| Physics | Vector calculations | Force components |
| Computer Graphics | Distance calculations | Game development |
Special Right Triangles
- 3-4-5 Triangle: 3² + 4² = 5² (9 + 16 = 25)
- 5-12-13 Triangle: 5² + 12² = 13² (25 + 144 = 169)
- 8-15-17 Triangle: 8² + 15² = 17² (64 + 225 = 289)
- 45°-45°-90° Triangle: Isosceles right triangle
- 30°-60°-90° Triangle: Special proportions
Distance Formula
- 2D Distance: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
- 3D Distance: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
- Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Triangle Properties
- Area: A = ½ × base × height
- Perimeter: P = a + b + c
- Right Triangle Area: A = ½ × leg₁ × leg₂
- Heron's Formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Note
This calculator provides accurate calculations for right triangles and distance measurements. For complex geometric problems, consider using specialized mathematical software.