Right Triangle Calculator

A right triangle is a triangle that contains one right angle (90 degrees). It is one of the most important geometric shapes. Defining Characteristic: One interior angle = exactly 90° (right angle) Components: 1. Hypotenuse (c): The longest side, opposite the right angle 2. Legs (a and b): The two sides that form the right angle 3. Angles: One 90° angle and two acute angles that sum to 90° The Pythagorean Theorem: Fundamental relationship: a² + b² = c² Where: • a, b = lengths of the two legs • c = length of the hypotenuse Derived formulas: • c = √(a² + b²) — Find hypotenuse • a = √(c² - b²) — Find one leg • b = √(c² - a²) — Find other leg Trigonometric Ratios: For angle θ (theta): • sin(θ) = opposite / hypotenuse • cos(θ) = adjacent / hypotenuse • tan(θ) = opposite / adjacent Special Right Triangles: 1. 45°-45°-90° Triangle (Isosceles Right Triangle): • Sides ratio: 1 : 1 : √2 • If legs = a, then hypotenuse = a√2 2. 30°-60°-90° Triangle: • Sides ratio: 1 : √3 : 2 • If shortest side = a, then other leg = a√3, hypotenuse = 2a Area and Perimeter: • Area = (1/2) × leg₁ × leg₂ = (1/2) × a × b • Perimeter = a + b + c (sum of all three sides) Key Properties: 1. Sum of angles = 180° (like all triangles) 2. The two acute angles are complementary (sum to 90°) 3. Hypotenuse is always the longest side 4. Only one right angle per triangle Applications: Construction and carpentry, navigation, surveying, physics (vector decomposition), computer graphics, roof design, ramp calculations, and distance measurements.