The Triangle Calculator is a comprehensive tool for solving various triangle problems. Given certain measurements (sides, angles, or a combination), it can calculate all remaining properties including missing sides, angles, area, perimeter, and other characteristics. Whether dealing with right triangles, equilateral, isosceles, or scalene triangles, this calculator applies the appropriate formulas and theorems.

Essential Triangle Formulas

Triangle Types

• By Sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (all sides different)
• By Angles: Right (one 90° angle), Acute (all angles less than 90°), Obtuse (one angle greater than 90°)
• Special Cases: 45-45-90 triangle, 30-60-90 triangle, and Pythagorean triples

Example 1: Right Triangle with Two Legs

Example 2: Using Heron's Formula (Three Sides Given)

Example 3: Using Law of Cosines (Two Sides and Included Angle)

Example 4: Equilateral Triangle

Example 5: Using Law of Sines (Two Angles and One Side)

Example 6: Special 30-60-90 Triangle

Essential Triangle Calculation Tips

• Triangle Inequality: The sum of any two sides must be greater than the third side (a + b > c). If not, no triangle exists
• Angle Sum Always 180°: All interior angles must add up to exactly 180°. Use this to find the third angle
• Choose the Right Formula: Use Pythagorean theorem for right triangles, Law of Cosines for SAS or SSS, Law of Sines for AAS or ASA
• Right Triangle Check: Test if c² = a² + b² (if true, it's a right triangle)
• Height vs Slant: For area calculations, use perpendicular height, not a slanted side
• Degrees vs Radians: Ensure your calculator is in the correct mode for angle calculations
• Heron's Formula: Use when you know all three sides but not the height
• Ambiguous Case (SSA): When given two sides and a non-included angle, there might be 0, 1, or 2 possible triangles
• Special Triangles: Memorize ratios for 30-60-90 (1:√3:2) and 45-45-90 (1:1:√2) triangles
• Unit Consistency: All length measurements must use the same units

Complete Formula Reference

Real-World Applications

• Construction: Roof pitch calculations, truss design, and structural bracing
• Navigation: Triangulation for GPS, determining positions using angles and distances
• Surveying: Measuring land, determining inaccessible distances, and property boundaries
• Architecture: Building design, staircase angles, and aesthetic proportions
• Engineering: Force analysis, stress calculations, and mechanical linkages
• Astronomy: Calculating distances to stars and celestial objects using parallax
• Computer Graphics: 3D rendering, polygon modeling, and texture mapping
• Physics: Vector decomposition, projectile motion, and wave interference
• Aviation: Flight path calculations, approach angles, and wind corrections
• Art and Design: Composition, perspective, and geometric patterns