Triangle Calculator
Enter any 3 of a triangle's sides and angles and solve the rest — area, perimeter, type — using the Law of Sines and Cosines. Free, runs in your browser.
Triangle Calculator
Sides
Angles (°)
Enter any 3 values (including at least one side). Angle A is opposite side a, and so on.
How to Use the Triangle Calculator
Enter what you know
Fill in any three of the six measurements — the three sides (a, b, c) and three angles (A, B, C) — including at least one side. Each angle sits opposite the side with the matching letter.
Let it solve the rest
The calculator finds every missing side and angle using the Law of Sines and the Law of Cosines, the standard tools for any triangle.
Read area and type
You also get the area, the perimeter, and the triangle's classification — right, acute, or obtuse, and equilateral, isosceles, or scalene.
Check it is possible
If no real triangle fits your numbers — angles over 180°, or a side too long — the calculator tells you instead of inventing an answer.
Solving Any Triangle
Three Pieces Determine the Whole
A triangle has six measurements — three sides and three angles — but they are not independent. Remarkably, knowing just three of them (as long as at least one is a side) is almost always enough to determine the other three exactly. That is the principle behind this calculator: give it any valid combination and it reconstructs the entire triangle. The two tools that make this possible are the Law of Sines, which relates each side to the sine of its opposite angle, and the Law of Cosines, which generalises the Pythagorean theorem to any angle — when the angle is 90° it reduces to a² + b² = c². Between them they cover every case students learn: three sides (SSS), two sides and the angle between them (SAS), two angles and a side (ASA or AAS). The calculator picks the right law automatically, so you never have to decide which formula applies; you just enter what you know.
This is why the triangle calculator is the hub of practical geometry. Triangulation — fixing a position by measuring angles and one known distance — is how surveyors map land, how GPS pins your location, how astronomers gauge stellar distances, and how navigation worked for centuries before satellites. Engineers and architects rely on triangles because they are the only rigid polygon: a triangle cannot be deformed without changing a side length, which is why bridges, roofs, and towers are full of triangular bracing. Anyone laying out a plot, cutting a rafter, or checking a survey is solving a triangle, and the same three-known-values logic applies whether the numbers are metres on a building site or light-years across the galaxy. Alongside the solved sides and angles, the calculator gives the area (via the reliable formula ½·a·b·sin C, which needs no height) and classifies the triangle so you understand its shape at a glance.
"Three measurements fix a triangle completely. That single fact — made usable by the Laws of Sines and Cosines — is how we survey land, navigate, and brace every bridge."
The One Tricky Case
Triangle-solving has one famous subtlety worth knowing: the "ambiguous case" (SSA), where you know two sides and an angle that is not between them. Depending on the numbers, that data can describe two different triangles, exactly one, or none at all — because the side opposite the known angle might swing to two positions. This is a genuine feature of geometry, not a flaw, and it is why entering at least one well-placed side and being clear about which angle is which matters. For the common, unambiguous cases — SSS, SAS, ASA, AAS — the answer is unique and the calculator delivers it instantly. It also guards the impossible inputs: angles that sum to 180° or more, or a "longest side" that is actually too short, return a clear message rather than a nonsense triangle. Whether you are checking trigonometry homework, working out a construction angle, or exploring a geometry problem, this calculator solves the triangle and explains its shape, privately and instantly in your browser.
10 Facts About Triangles
Knowing three of six measurements (incl. one side) fixes the whole triangle.
The angles always sum to 180° in a flat triangle.
The Law of Sines relates each side to the sine of its opposite angle.
The Law of Cosines generalises Pythagoras to any angle.
A triangle is the only rigid polygon — it can't be deformed.
That rigidity is why bridges and roofs use triangular bracing.
Triangulation underlies surveying, GPS, and astronomy.
Area = ½·a·b·sin C — no height needed.
The SSA "ambiguous case" can give two, one, or no triangles.
Triangles classify by angle (right/acute/obtuse) and sides.
Frequently Asked Questions
- Three, and at least one must be a side. Common valid combinations are three sides (SSS), two sides and the included angle (SAS), and two angles with any side (ASA or AAS). Knowing three angles alone is not enough, because that fixes the shape but not the size.
- They are the two rules for any triangle. The Law of Sines says each side divided by the sine of its opposite angle is the same ratio; the Law of Cosines relates one side to the other two and the angle between them. The calculator picks whichever is needed automatically, so you do not have to choose.
- Each angle is opposite the side with the matching letter: angle A is opposite side a, angle B opposite side b, angle C opposite side c. Keeping this pairing straight is the key to entering your data correctly, and the labels in the calculator follow this standard convention.
- Using the formula ½·a·b·sin C — half the product of two sides times the sine of the angle between them. This works for any triangle and needs no separate height measurement. When all three sides are known, it is equivalent to Heron's formula. The calculator computes it automatically.
- When you know two sides and an angle not between them, the data can sometimes fit two different triangles, just one, or none — the side opposite the known angle may reach two positions. This is a real property of geometry. For the unambiguous cases (SSS, SAS, ASA, AAS) the answer is unique, which is where the calculator is most reliable.
- Because the numbers you entered cannot form a real triangle — for example the angles add up to 180° or more, or one side is too long for the others to reach. Rather than inventing an answer, the calculator flags it so you can correct the input. The triangle inequality and the 180° angle sum are the rules being checked.
- Two ways. By angles: right (one 90° angle), acute (all under 90°), or obtuse (one over 90°). By sides: equilateral (all equal), isosceles (two equal), or scalene (all different). The calculator reports both, so you get a complete picture of the triangle's shape.
- Yes — just enter the 90° angle as one of your three values, along with two more measurements. The Law of Cosines reduces to the Pythagorean theorem when an angle is 90°, so right triangles are handled like any other. For a quick two-sides-of-a-right-triangle job, the Pythagorean Theorem Calculator is even faster.
- Yes. The calculation runs in your browser — nothing is uploaded, stored, or logged — and it works offline once loaded.
- Completely free, with no account, sign-up, or limit. It runs entirely in your browser and collects no data. Use it as often as you like.
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