Calculate Triangle Properties with Trigonometry & Geometry Formulas

Calculate triangle area, perimeter & angles using various inputs. Solve for missing measurements with trigonometry formulas. Free geometry calculator for homework & construction projects.

Calculate triangle area, perimeter, angles, and other properties using various input combinations. This comprehensive triangle calculator works with side lengths, angles, height, and base measurements to solve any triangle problem.

How to Use This Triangle Calculator

This comprehensive calculator can solve triangles using various input methods:

Triangle Properties Calculated

The calculator provides complete triangle information including:

Triangle Formulas Used

Area Formulas:

Law of Cosines: c² = a² + b² - 2ab cos(C)

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Note: For a valid triangle, the sum of any two sides must be greater than the third side, and the sum of all angles must equal 180°. The calculator will validate inputs and provide error messages for invalid triangles.

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Frequently Asked Questions

What information do I need to calculate triangle properties?

Triangle calculations require different combinations of known measurements: three sides (SSS), two sides and included angle (SAS), two angles and one side (AAS/ASA), or base and height for area. With three sides, you can find all angles and area. With two sides and included angle, you can find the third side and remaining angles. Right triangles need only two sides (Pythagorean theorem) or one side and one angle (trigonometry). Insufficient information (like two angles only) cannot determine unique triangle properties.

How do I calculate triangle area with different given measurements?

Multiple formulas calculate triangle area depending on known values: Base × Height ÷ 2 (most basic), Heron's formula when three sides are known: A = √(s(s-a)(s-b)(s-c)) where s = semi-perimeter, SAS formula: A = ½ab sin(C) when two sides and included angle known, ASA formula: A = (a² sin(B) sin(C))/(2 sin(A)). Right triangles use A = ½ × leg₁ × leg₂. Choose appropriate formula based on available measurements for most efficient calculation.

How do I find missing angles when I know the sides of a triangle?

Use Law of Cosines to find angles when three sides are known: cos(A) = (b² + c² - a²)/(2bc), then A = arccos of that result. Repeat for other angles, or use fact that angles sum to 180°. For right triangles, use inverse trigonometric functions: if opposite and adjacent sides known, angle = arctan(opposite/adjacent). Law of Sines also works: sin(A)/a = sin(B)/b = sin(C)/c. Calculator will typically require inverse trig functions (arcsin, arccos, arctan) to convert ratios back to angles.

What's the difference between different types of triangles?

Triangles classify by sides: equilateral (all sides equal, all angles 60°), isosceles (two sides equal, two angles equal), scalene (no equal sides or angles). By angles: acute (all angles under 90°), right (one 90° angle), obtuse (one angle over 90°). Properties differ: right triangles use Pythagorean theorem (a² + b² = c²), equilateral triangles have area = (√3/4) × side², isosceles triangles have line of symmetry. Understanding triangle type helps choose appropriate formulas and solution methods.

What are common real-world applications of triangle calculations?

Triangle calculations appear in construction (roof angles, structural supports), navigation (triangulation for GPS), surveying (measuring distances and elevations), engineering (force analysis, bridge design), architecture (calculating dimensions, angles), astronomy (measuring stellar distances), and art (perspective, composition). Examples: finding roof pitch angle, calculating ladder placement angle for safety, determining triangle lot area, measuring building heights using shadows and angles. Understanding triangles helps solve practical geometric problems in many fields requiring spatial analysis and measurement.