Triangle Angle Calculator
Enter any three values that define a triangle (at least one must be a side length) and the calculator will find every missing angle, side, area, perimeter, inradius, and circumradius. It works for all triangle types: SSS (three sides), SAS (two sides and included angle), ASA, AAS, and SSA. Results update instantly as you type, and the step-by-step panel shows the exact formulas used.
How to use this triangle angle calculator
Enter any three values that define your triangle, making sure at least one is a side length. You can leave up to three fields blank. The calculator detects the case automatically (SSS, SAS, ASA, AAS, or SSA) and fills in all missing angles and sides. For SSA (two sides and a non-included angle), the calculator returns the primary solution; if the geometry produces two valid triangles, the ambiguous case note in the FAQs explains how to handle that. All angles are in degrees.
Formulas used
Law of cosines (SSS and SAS): a² = b² + c² - 2bc cos(A). Rearranged to find angles: cos(A) = (b² + c² - a²) / (2bc). Law of sines (AAS, ASA, SSA): a/sin(A) = b/sin(B) = c/sin(C). Angle sum theorem (all cases): A + B + C = 180 degrees. Area by Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Inradius: r = Area / s. Circumradius: R = (abc) / (4 Area).
Triangle types and what they mean
Acute triangle: all angles below 90 degrees. Obtuse triangle: one angle above 90 degrees. Right triangle: one angle exactly 90 degrees, governed by the Pythagorean theorem (a² + b² = c² where c is the hypotenuse). Equilateral triangle: all three sides equal, all angles 60 degrees. Isosceles triangle: two sides equal, two base angles equal. Scalene triangle: all sides and angles different.
The ambiguous case (SSA)
When you know two sides and a non-included angle, there can be zero, one, or two valid triangles. If sin(B) > 1 there is no solution. If sin(B) = 1 there is exactly one right triangle. If sin(B) < 1 there are generally two possible triangles (B and 180 - B), unless the given angle is already obtuse. This calculator returns the acute-angle solution first. If you need the obtuse alternative, subtract the returned B from 180 degrees and re-check whether the angles still sum to 180.
Triangle congruence cases
| Case | Known values | Method | Unique solution? |
|---|---|---|---|
| SSS | 3 sides | Law of cosines | Yes |
| SAS | 2 sides + included angle | Law of cosines | Yes |
| ASA | 2 angles + included side | Angle sum + law of sines | Yes |
| AAS | 2 angles + non-included side | Angle sum + law of sines | Yes |
| SSA | 2 sides + non-included angle | Law of sines | Not always (ambiguous case) |
| AAA | 3 angles only | Not enough info - shape only, not size | No |
Which combination of known values uniquely determines a triangle.
Frequently asked questions
What information do I need to solve a triangle?
You need at least three values, and at least one must be a side length. Three angles alone (AAA) only fix the shape, not the size. The five solvable cases are: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle, which can be ambiguous).
How do I find a missing angle in a triangle?
If you know the other two angles, subtract their sum from 180 degrees (angle sum theorem). If you know all three sides, use the law of cosines: cos(A) = (b² + c² - a²) / (2bc). If you know two sides and an angle, use either the law of cosines (SAS) or law of sines (SSA) depending on which angle is given.
What is the law of cosines?
The law of cosines generalises the Pythagorean theorem to any triangle: a² = b² + c² - 2bc cos(A). When angle A is 90 degrees, cos(A) = 0 and the formula reduces to a² = b² + c². It is used to find a missing side (SAS case) or a missing angle (SSS case).
What is the law of sines?
The law of sines states that each side of a triangle divided by the sine of the opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This ratio equals the diameter of the circumscribed circle (2R). The law of sines is used for ASA, AAS, and SSA cases.
Can a triangle have two right angles?
No. The three angles must sum to exactly 180 degrees. Two right angles would already use 180 degrees, leaving nothing for the third angle, which must be positive. A valid triangle can have at most one right angle or one obtuse angle.
What is the inradius and why does it matter?
The inradius (r) is the radius of the largest circle that fits entirely inside the triangle, tangent to all three sides. It equals the area divided by the semi-perimeter (r = Area / s). It is used in geometry, engineering tolerances, and inscribed-circle problems.
What is the circumradius?
The circumradius (R) is the radius of the circle that passes through all three vertices of the triangle. It is calculated as R = (abc) / (4 Area). For a right triangle, the circumradius equals half the hypotenuse.