Triangle Calculator

Triangle Calculator: Solving Any Triangle

A triangle is defined by six elements: three sides and three angles. Given any three of these (with at least one side), all remaining elements can be determined. This calculator supports four fundamental configurations: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), and AAS (two angles and a non-included side). Each uses different formulas from trigonometry to solve the triangle completely, and the calculator provides a visual diagram that updates in real time.

Key Formulas for Triangle Solving

Heron's Formula (SSS for area):
  s = (a + b + c) / 2
  Area = sqrt[s(s-a)(s-b)(s-c)]

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

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

Area = (1/2) * a * b * sin(C)

Circumradius: R = a / (2*sin(A))
Inradius: r = Area / s

Triangle Types and Properties

Triangles are classified by sides (equilateral: all equal, isosceles: two equal, scalene: all different) and by angles (acute: all less than 90°, right: one equals 90°, obtuse: one greater than 90°). Every triangle has an angle sum of exactly 180°. The longest side is always opposite the largest angle. The triangle inequality theorem requires that the sum of any two sides must exceed the third side. The circumradius R is the radius of the circumscribed circle through all vertices, while the inradius r is the radius of the inscribed circle tangent to all three sides.

Which Mode Should I Use?

SSS: You know all three side lengths. Uses Law of Cosines for angles, Heron's formula for area. SAS: You know two sides and the angle between them. Uses Law of Cosines to find the third side. ASA: You know two angles and the side between them. The third angle is 180 minus the other two, then Law of Sines finds remaining sides. AAS: You know two angles and a side not between them. Same approach as ASA after computing the third angle.

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