Triangle Calculator

What is a Triangle Calculator?

A triangle calculator is a powerful online geometry tool that computes all key properties of a triangle from a minimal set of known measurements. Whether you know three sides, two sides and an angle, or two angles and a side, this calculator uses fundamental triangle theorems and trigonometry to solve for all remaining unknowns.

The calculator determines side lengths, interior angles, area, perimeter, heights (altitudes), and automatically classifies the triangle type (right, acute, obtuse, equilateral, isosceles, or scalene). This comprehensive approach saves time and reduces errors in multi-step geometric calculations.

Perfect for students, teachers, engineers, architects, surveyors, and designers, the triangle calculator provides instant, accurate results for homework problems, construction layouts, structural analysis, CAD drawings, and land area estimation.

How to Use the Triangle Calculator

1. Select your input mode based on what measurements you know: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), AAS (two angles and non-included side), or SSA (two sides and non-included angle).
2. Enter the known values in the appropriate fields. For sides, enter positive numbers. For angles, enter values in degrees between 0 and 180.
3. Click the 'Calculate Triangle' button to compute all triangle properties.
4. Review the comprehensive results including all sides, angles, area, perimeter, heights, and triangle classification.
5. Use the results for your specific application, whether it's homework verification, construction planning, or design work. The calculator validates inputs and alerts you to impossible triangle configurations.

Key Features

• Multiple Input Modes: Supports SSS, SAS, ASA, AAS, and SSA configurations, allowing you to solve triangles from any valid combination of known measurements.
• Comprehensive Results: Calculates all sides, angles, area, perimeter, and heights (altitudes) in a single operation.
• Triangle Classification: Automatically identifies triangle type by angle (right, acute, obtuse) and by sides (equilateral, isosceles, scalene).
• Input Validation: Checks triangle inequality and angle sum constraints to ensure only valid triangles are computed.
• High Precision: Uses advanced trigonometric algorithms and maintains precision throughout multi-step calculations.
• Educational Value: Helps students understand relationships between triangle properties and verify manual calculations.

Mathematical Basis

The triangle calculator implements fundamental theorems of Euclidean geometry and trigonometry to solve for unknown triangle properties:

Triangle Inequality Theorem

For any triangle with sides a, b, and c, the sum of any two sides must be greater than the third side: a + b > c, a + c > b, and b + c > a. The calculator validates this before computing results.

Pythagorean Theorem

For right triangles, the square of the hypotenuse equals the sum of squares of the other two sides: c² = a² + b². This is used to identify and solve right triangles.

Law of Sines

For any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This is essential for solving ASA, AAS, and SSA cases.

Law of Cosines

Generalizes the Pythagorean theorem for any triangle: c² = a² + b² - 2ab·cos(C). Used to find unknown sides in SAS cases and unknown angles in SSS cases.

Heron's Formula

Calculates triangle area from three sides: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter. The calculator uses numerically stable variants for extreme triangle shapes.

Best Practices & Tips

• Provide Sufficient Data: You must supply at least three independent measurements that uniquely determine a triangle. Inconsistent or insufficient inputs cannot produce valid results.
• Use Consistent Angle Units: This calculator expects all angles in degrees. Ensure you don't mix degrees and radians, and keep angle values between 0° and 180°.
• Verify Triangle Inequality: When entering three sides, make sure they satisfy the triangle inequality. The calculator will flag impossible triangles, but pre-checking saves time.
• Understand SSA Ambiguity: The SSA (two sides and non-included angle) case can sometimes have two valid solutions or no solution. The calculator returns one valid solution when it exists.
• Maintain Precision: Avoid rounding intermediate results when doing manual calculations. The calculator maintains full precision throughout all computations.
• Cross-Check Critical Results: For engineering, construction, or safety-critical applications, verify calculator results independently and consult professionals when necessary.

Common Use Cases

• Education & Homework: Students use the calculator to verify geometry homework, understand triangle relationships, and check manual calculations for accuracy.
• Construction & Carpentry: Builders and carpenters calculate roof angles, truss dimensions, and structural triangulation for accurate construction layouts.
• Land Surveying: Surveyors determine property boundaries, plot areas, and distances using triangulation methods with measured angles and sides.
• Engineering Design: Engineers analyze structural loads, design triangular components, and verify geometric constraints in mechanical and civil projects.
• CAD & Technical Drawing: Designers calculate precise dimensions for triangular elements in architectural plans, mechanical drawings, and product designs.
• Navigation & Mapping: Navigators and cartographers use triangle calculations for distance estimation, position fixing, and route planning (in flat-plane approximations).

Frequently Asked Questions

References & Further Reading