Enter any three of sides a, b, c and angles A, B, C — including at least one side — and this calculator fills in the rest. It covers SSS, SAS, ASA/AAS, and the SSA ambiguous case (zero, one, or two triangles), then reports area, perimeter, heights, medians, inradius, circumradius, and vertex coordinates.
Given three parts
| Case | Known | Method |
|---|---|---|
| SSS | Three sides | Law of cosines for each angle |
| SAS | Two sides and the included angle | Law of cosines for the third side |
| ASA / AAS | Two angles and any side | Angle sum, then law of sines |
| SSA | Two sides and a non-included angle | Law of sines; 0, 1, or 2 triangles |
The sum of any two sides must exceed the third. Interior angles must be between 0° and 180° and add to 180°.
Examples
Default equilateral (C = 60°, a = 1, b = 1)
SAS with two equal sides and a 60° included angle is equilateral: c = 1, every angle is 60°, area 0.43301, perimeter 3, heights and medians 0.86603, inradius 0.28868, circumradius 0.57735.
SAS a = 7, b = 9, C = 30°
c ≈ 4.56955, area 15.75, ∠A ≈ 49.991°, ∠B ≈ 100.009°. One obtuse angle makes this an obtuse scalene triangle.
SSS 3-4-5 right triangle
Sides 3, 4, 5 satisfy 3² + 4² = 5², so ∠C is 90°, area 6, inradius 1, and circumradius 2.5 (half the hypotenuse).
SSA two solutions (A = 30°, a = 5, b = 8)
Possible 1: c ≈ 9.9282, ∠B ≈ 53.13°, ∠C ≈ 96.87°, area ≈ 19.856. Possible 2: c ≈ 3.9282, ∠B ≈ 126.87°, ∠C ≈ 23.13°, area ≈ 7.856.
Formula
Side a is opposite angle A, b opposite B, and c opposite C. The law of cosines handles SSS and SAS:
c² = a² + b² − 2ab · cos(C)
The law of sines handles ASA, AAS, and SSA:
a / sin(A) = b / sin(B) = c / sin(C)
Area from two sides and the included angle, or from Heron's formula once all three sides are known:
area = (1/2) · a · b · sin(C)
area = √(s(s − a)(s − b)(s − c))
s = (a + b + c) / 2
Heights, medians, and the two radii follow from the solved triangle:
ha = 2 · area / a
ma = (1/2) · √(2b² + 2c² − a²)
r = area / s
R = a / (2 sin A)
Vertices are placed with A at (0, 0) and B at (c, 0), matching the usual diagram. The centroid, incenter, and circumcenter are computed from those coordinates.
SSA is the ambiguous case: after finding sin B = b sin(A) / a, both B and 180° − B can be valid. Both solutions are listed when they form a triangle.
Solution limits
Inputs must define a nondegenerate Euclidean triangle. In the SSA case, the same measurements can produce zero, one, or two triangles; review every returned solution. Rounded measurements can move a case across a triangle-inequality or angle-sum boundary, and the calculator does not handle spherical triangles.