Pythagoras Theorem Calculator & Right Triangle Solver

Pythagoras Theorem Guide

Pythagoras Theorem

In a right triangle: a² + b² = c². Where c is the hypotenuse (longest side, opposite the right angle). Finding the hypotenuse: c = √(a² + b²). Finding a shorter side: a = √(c² − b²). The 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5². This is the simplest Pythagorean triple. Other triples: 5-12-13, 8-15-17, 7-24-25. Distance formula in coordinate geometry: distance between (x₁,y₁) and (x₂,y₂) = √((x₂−x₁)² + (y₂−y₁)²) is Pythagoras in 2D. 3D distance: d = √(a² + b² + c²).

Trigonometry in Right Triangles

SOH CAH TOA: sin(θ) = Opposite/Hypotenuse. cos(θ) = Adjacent/Hypotenuse. tan(θ) = Opposite/Adjacent. Finding angles: angle A = arcsin(a/c) = arccos(b/c) = arctan(a/b). All angles of a right triangle: one is 90°. The other two sum to 90° (complementary angles). If angle A = 30°: angle B = 60°. Special triangles: 30-60-90 (sides in ratio 1:√3:2). 45-45-90 (sides in ratio 1:1:√2). These give exact values: sin(30°) = 0.5, cos(60°) = 0.5, tan(45°) = 1.

Pythagoras in 3D

Length of space diagonal of a cuboid (box): d = √(l² + w² + h²). This is applying Pythagoras twice: first to the base (√(l² + w²)), then using that diagonal and the height. Example: box 3m × 4m × 5m. Base diagonal = √(9+16) = 5m. Space diagonal = √(25+25) = √50 = 7.07m. Maximum length of item fitting in the box: 7.07m. Construction uses: checking square corners: a 3-4-5 rope gives a perfect right angle. Surveying: horizontal distances calculated from slant measurements. Navigation: calculating s

Pythagorean Triples

Pythagorean triples are integer solutions to a² + b² = c². Generator formula: a = m² − n², b = 2mn, c = m² + n² for any positive integers m > n. For m=2, n=1: a=3, b=4, c=5. For m=3, n=2: a=5, b=12, c=13. For m=4, n=3: a=7, b=24, c=25. For m=5, n=2: a=21, b=20, c=29. Infinite families: multiplying any triple by a constant gives another triple. 3-4-5, 6-8-10, 9-12-15... Fermat's Last Theorem (proved 1994 by Andrew Wiles): no integer solutions exist for aⁿ + bⁿ = cⁿ when n > 2. The Pythagorean tri