Complete triangle solver for sides, angles, area, and perimeter. Supports all triangle types with law of sines, law of cosines, and Heron's formula calculations.
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry and serves as the foundation for many mathematical concepts including trigonometry, construction, engineering, and design.
Understanding triangle properties and calculations is essential for solving geometric problems, calculating areas and perimeters, determining angles and sides, and applying trigonometric principles in real-world scenarios.
Where s = (a+b+c)/2 (semi-perimeter)
Used to find unknown sides or angles
Used when two sides and included angle are known
Property | Formula/Rule | Application | Example |
---|---|---|---|
Angle Sum | A + B + C = 180° | Find missing angle | If A=60°, B=70°, then C=50° |
Perimeter | P = a + b + c | Total boundary length | P = 3 + 4 + 5 = 12 |
Triangle Inequality | a + b > c | Validate triangle existence | 3 + 4 > 5 ✓ Valid |
Pythagorean Theorem | a² + b² = c² (right triangles) | Right triangle identification | 3² + 4² = 5² ✓ Right triangle |
Median Length | m = ½√(2a² + 2b² - c²) | Line from vertex to midpoint | Geometric constructions |
Altitude | h = 2×Area/base | Height calculations | Design and engineering |
SSS (Side-Side-Side): When all three sides are known, use the law of cosines to find angles, then calculate area using Heron's formula.
SAS (Side-Angle-Side): When two sides and the included angle are known, use the law of cosines to find the third side, then use law of sines for remaining angles.
ASA (Angle-Side-Angle): When two angles and the included side are known, find the third angle using angle sum property, then use law of sines to find remaining sides.
AAS (Angle-Angle-Side): When two angles and a non-included side are known, find the third angle, then use law of sines to find remaining sides.
SSA (Side-Side-Angle): Ambiguous case that may have 0, 1, or 2 solutions. Requires careful analysis using law of sines.
Triangle Type | Side Ratios | Angle Measures | Applications |
---|---|---|---|
30-60-90 | 1 : √3 : 2 | 30°, 60°, 90° | Construction, design |
45-45-90 | 1 : 1 : √2 | 45°, 45°, 90° | Square diagonals |
3-4-5 | 3 : 4 : 5 | Various angles | Carpentry, surveying |
5-12-13 | 5 : 12 : 13 | Various angles | Pythagorean triple |
Equilateral | 1 : 1 : 1 | 60°, 60°, 60° | Regular patterns |
Architecture and Construction: Triangles provide structural stability in buildings, bridges, and towers. Roof trusses, support beams, and architectural frameworks rely on triangular geometry.
Navigation and Surveying: Triangulation methods use triangle calculations to determine distances and positions in GPS systems, marine navigation, and land surveying.
Engineering Design: Mechanical engineering uses triangle calculations for force analysis, stress distribution, and component design in machinery and structures.
Computer Graphics: 3D modeling and game development extensively use triangular meshes and triangle calculations for rendering and collision detection.
Verify Triangle Validity: Always check that the triangle inequality holds: the sum of any two sides must be greater than the third side.
Choose Appropriate Method: Select the most suitable formula based on known values. Use law of sines for angle-side relationships and law of cosines for side-side-side problems.
Check Angle Sum: Verify that calculated angles sum to 180° (within rounding tolerance) to ensure accuracy.
Consider Precision: Be aware of rounding errors in calculations, especially when working with very small or very large triangles.