What this calculator does
The Triangle Theorems Calculator solves a complete triangle from one of six classic input combinations: AAA (three angles), AAS (two angles and a side), ASA (two angles and the included side), ASS/SSA (an angle and two sides, the ambiguous case), SAS (two sides and the included angle), and SSS (three sides). It uses standard triangle labeling where angle A is opposite side a, angle B opposite side b, and angle C opposite side c.
How to use it
Pick the theorem that matches the information you have from the "Calculate:" dropdown, then enter the three relevant values. Choose whether your angles are in degrees or radians, optionally pick a length unit (purely a display label - all sides must share the same unit), and set the number of significant figures. The result lists all three angles, all three sides, plus perimeter P, semi-perimeter s, area K, inradius r, and circumradius R.
The formulas explained
The angle sum rule (\(A + B + C = 180\) degrees) fills in a missing angle. The Law of Sines, $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,$$ finds unknown sides when an angle and its opposite side are known. The Law of Cosines, $$a^2 = b^2 + c^2 - 2bc\cos A,$$ handles SAS and SSS. Once all sides and angles are known, Heron's formula gives area $$K = \sqrt{s(s-a)(s-b)(s-c)},$$ the inradius is \(r = K/s\), and the circumradius is \(R = abc/(4K)\).
Worked example (ASA)
Given \(A = 60\degree\), included side \(c = 10\), \(B = 50\degree\): first $$C = 180 - 60 - 50 = 70\degree.$$ By the Law of Sines, $$a = \frac{10\cdot\sin 60\degree}{\sin 70\degree} = 9.21595$$ and $$b = \frac{10\cdot\sin 50\degree}{\sin 70\degree} = 8.15205.$$ Then \(P = 27.3680\), \(s = 13.6840\), \(K = 35.2912\), \(r = 2.57902\), and \(R = 5.32200\).
FAQ
Why does AAA not give side lengths? Three angles fix only the shape, not the size - there are infinitely many similar triangles, so sides, perimeter, area and radii are undefined.
What is the ambiguous (SSA) case? When you know an angle and two sides where the angle is not included, there can be zero, one, or two valid triangles. This tool reports the primary solution and flags impossible cases.
Does it convert between length units? No - the length unit is just a label appended to outputs. Enter all sides in the same unit; the area is reported in that unit squared.
