Solving a Triangle
A triangle has six measurements — three sides and three angles. "Solving" one means finding all six from a known subset. Three values are almost always enough, with one important exception: three angles fix the shape but not the size, since any scaled copy has the same angles. At least one side is always required.
The Valid Combinations
| Known | Name | Method | Unique? |
|---|---|---|---|
| 3 sides | SSS | Law of cosines | Yes, if the triangle inequality holds |
| 2 sides + included angle | SAS | Law of cosines | Yes |
| 2 angles + any side | ASA / AAS | Law of sines | Yes |
| 2 sides + non-included angle | SSA | Law of sines | No — ambiguous |
| 3 angles | AAA | — | No — shape only |
The Two Laws
The law of cosines generalises the Pythagorean theorem to any triangle:
a2 = b2 + c2 − 2bc · cos(A)
When A = 90°, cos(A) = 0 and the final term vanishes, leaving a2 = b2 + c2. Pythagoras is the right-angled special case.
The law of sines states that each side is proportional to the sine of its opposite angle:
a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C) = 2R
where R is the radius of the circumscribed circle — a fact that comes free with the law.
The Ambiguous Case (SSA)
Given two sides and an angle not between them, there can be two different triangles, one, or none. The reason is that sin(θ) = sin(180° − θ), so the law of sines cannot distinguish an acute answer from its obtuse partner.
| Condition (given angle A, sides a and b) | Number of triangles |
|---|---|
| a < b · sin(A) | None — side a is too short to reach |
| a = b · sin(A) | One — a right triangle |
| b · sin(A) < a < b | Two — the ambiguous case |
| a ≥ b | One |
The calculator returns the acute solution, which is the conventional choice. If your problem context allows an obtuse angle, subtract the reported angle from 180° to obtain the second triangle.
The Triangle Inequality
Three lengths form a triangle only if each is shorter than the sum of the other two:
a + b > c a + c > b b + c > a
Lengths 2, 3 and 9 cannot close: the two shorter sides together fall short of the longest. The calculator checks this before attempting to solve.
Area Formulas
| Known | Formula |
|---|---|
| Base and height | Area = ½ × base × height |
| Three sides | Heron: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 |
| Two sides + included angle | Area = ½ × a × b × sin(C) |
| Coordinates | Shoelace: Area = ½|x1(y2−y3) + x2(y3−y1) + x3(y1−y2)| |
This calculator uses Heron's formula, since it always has all three sides by the time the area is computed.
Classifying Triangles
| By sides | Definition | By angles | Definition |
|---|---|---|---|
| Equilateral | All three sides equal; all angles 60° | Acute | All angles under 90° |
| Isosceles | Two sides equal; the opposite angles are equal | Right | One angle exactly 90° |
| Scalene | All sides different | Obtuse | One angle over 90° |
Common Right Triangles
Certain side ratios recur often enough to be worth recognising.
| Ratio | Angles | Note |
|---|---|---|
| 3 : 4 : 5 | 36.87°, 53.13°, 90° | The most common Pythagorean triple |
| 5 : 12 : 13 | 22.62°, 67.38°, 90° | Pythagorean triple |
| 8 : 15 : 17 | 28.07°, 61.93°, 90° | Pythagorean triple |
| 1 : 1 : √2 | 45°, 45°, 90° | Half a square |
| 1 : √3 : 2 | 30°, 60°, 90° | Half an equilateral triangle |
The 3-4-5 triangle is why builders can square a corner with nothing but a tape measure: mark 3 units along one edge, 4 along the other, and adjust until the diagonal reads exactly 5.
Circumradius and Inradius
Every triangle has a unique circle through all three vertices (the circumcircle) and a unique circle tangent to all three sides (the incircle).
R = abc ÷ 4 × Area r = Area ÷ s
For a right triangle, R is exactly half the hypotenuse — the right angle sits on the circle, by Thales' theorem.
Frequently Asked Questions
Why can't three angles determine the triangle?
Because angles describe shape, not size. Two triangles with identical angles are similar — one is a scaled copy of the other — and there are infinitely many such copies.
Are the angles in degrees or radians?
Degrees. Multiply by π/180 to convert to radians if you need them.
Why did I get "no triangle exists"?
Either the three sides violate the triangle inequality, the angles do not sum to 180°, or in an SSA case the side opposite the given angle is too short to reach the third vertex.
How do I find the area from coordinates?
Compute the three side lengths with the distance formula and enter them here, or apply the shoelace formula in the table above directly.