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Trigonometry & Geometry
Triangle Area Calculator (All Methods)
Compute the exact area of any triangle using Base & Height, Heron's Formula (3 sides), or SAS Trigonometry (2 sides and angle) with step-by-step mathematical proofs.
units
units
Calculated Triangle Area
30.0000
Perimeter: 0.0000 units
Understanding Triangle Area Formulas
A triangle is a fundamental polygon with three vertices and three edges. The area represents the total 2D space enclosed within those three boundaries.
Core Geometric Triangle Formulas
Standard (Base & Height): Area = ½ × base × height
Heron's Formula: s = (a + b + c) / 2, Area = √[s(s - a)(s - b)(s - c)]
Trigonometric (SAS): Area = ½ × a × b × sin(γ)
Triangle Inequality Theorem
When calculating area from three sides (Heron's method), the three lengths must satisfy the Triangle Inequality Theorem: the sum of the lengths of any two sides must be strictly greater than the third side (a + b > c, a + c > b, b + c > a). If this condition is violated, the shape cannot close and forms an impossible geometry.
Frequently Asked Questions
Yes, Heron's formula works for any valid triangle (equilateral, isosceles, scalene, or right). For a 3-4-5 right triangle, Heron's gives √[6(3)(2)(1)] = √36 = 6, identical to ½ × 3 × 4 = 6.
This calculator accepts angles in degrees (°). For angles between 0° and 180°, standard degrees are converted internally to radians using (angle × π / 180).
Educational Disclaimer: Local JavaScript math execution uses IEEE-754 double precision standard.