Triangle calculator: solve any triangle from three values
This triangle calculator finds every missing side and angle of a triangle from three known values. Pick what you know — three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA), two angles and another side (AAS), or two sides and an angle not between them (SSA) — and it returns all three sides, all three angles, the area, perimeter, heights, medians, inradius and circumradius, with the working and a scale drawing.
It handles the awkward SSA case honestly: when the numbers allow two different triangles it shows both and lets you switch between them, and when no triangle exists it says why. Angles can be entered and shown in degrees or radians. Everything is computed in your browser, with nothing uploaded.
How to use the Triangle Calculator
- Choose the combination you know (SSS, SAS, ASA, AAS or SSA) from the first menu.
- Enter the three values. Side a is opposite angle A, side b opposite B and side c opposite C; the labels and the hint change with the mode.
- Choose degrees or radians and, optionally, a unit of length.
- Read the triangle type, the solved sides and angles, and the working underneath.
- In the ambiguous SSA case, switch between Triangle 1 and Triangle 2 to see both solutions.
Triangle Calculator features
All five cases
SSS, SAS, ASA, AAS and SSA, each with its own input labels and a worked example loaded.
Ambiguous case handled
SSA gives two, one or no triangles depending on the height b·sin A, and the page tells you which.
Area two ways
Area from ½ab·sin C is cross-checked against Heron’s formula so you can see they agree.
Beyond the sides
Heights, medians, inradius, circumradius, perimeter and a classification such as acute scalene or right isosceles.
Scale drawing
The triangle is drawn in true proportion with sides and angles labelled.
Which formula solves which case
Two laws do almost all the work. The law of cosines, c² = a² + b² − 2ab·cos C, finds a side from two sides and the angle between them, or an angle when all three sides are known. The law of sines, a / sin A = b / sin B = c / sin C, links each side to its opposite angle and solves the ASA, AAS and SSA cases. The three angles of a triangle always add to 180° (π radians), which supplies the last angle once two are known.
The area follows from any two sides and their included angle, ½ab·sin C. When only the sides are known, Heron’s formula gives the same number: with s = (a + b + c) / 2, the area is √(s(s − a)(s − b)(s − c)). The inradius is the area divided by s, and the circumradius is a / (2 sin A).
| You know | Name | Main formula |
|---|---|---|
| 3 sides | SSS | Law of cosines for the angles |
| 2 sides + included angle | SAS | Law of cosines, then law of sines |
| 2 angles + included side | ASA | Angle sum, then law of sines |
| 2 angles + other side | AAS | Angle sum, then law of sines |
| 2 sides + other angle | SSA | Law of sines (0, 1 or 2 answers) |
The SSA ambiguous case
With two sides and a non-included angle, the side opposite the known angle may swing to meet the base in two places. Compare side a with the height h = b·sin A. If a is shorter than h, it cannot reach the base and there is no triangle. If a equals h there is one right triangle. If h is less than a and a is less than b, there are two triangles whose angles B are supplementary (they add to 180°). If a is at least b, there is just one. For an obtuse angle A only a > b works.
Example: a = 6, b = 8 and A = 30° give h = 4, so a lies between the height and b. One triangle has B ≈ 41.81°, the other B ≈ 138.19°. For a right triangle with two legs or a leg and hypotenuse, the Pythagorean theorem calculator is quicker, and the area calculator covers other shapes.
Triangle types and limits
By sides a triangle is equilateral (all equal), isosceles (two equal) or scalene (none equal). By angles it is acute (all under 90°), right (one of 90°) or obtuse (one over 90°). The tool applies a tiny tolerance when deciding equality so values like 3-4-5 are recognised as right-angled despite rounding.
Results are shown to six significant decimals and are only as exact as the numbers you enter. Measured real-world sides carry measurement error, so near-degenerate triangles (one angle close to 0° or 180°) can change a lot from small input changes. This is a maths tool, not a surveying instrument.
Triangle Calculator FAQ
How do I find the angles of a triangle from three sides?
Choose SSS, enter the sides and read the angles. The tool uses the law of cosines, cos A = (b² + c² − a²) / 2bc, and the angle sum for the third angle. The sides must satisfy the triangle inequality: each is shorter than the other two combined.
How do I find the area of a triangle?
With two sides and the angle between them the area is ½ab·sin C. With three sides use Heron’s formula, √(s(s − a)(s − b)(s − c)). The calculator shows both and they agree.
Can this be used as a right triangle calculator?
Yes. Enter any combination that fits a right triangle, such as SAS with a 90° angle or SSS with 3, 4, 5, and the type shows as right. For a dedicated leg and hypotenuse tool with triples and exact roots, use the Pythagorean theorem calculator.
Why does SSA sometimes give two triangles?
The side opposite the known angle can meet the base at two points when it is longer than the height b·sin A but shorter than side b. The two triangles have supplementary angles B and B′ = 180° − B.
Why does it say no triangle exists?
Either the sides break the triangle inequality, the angles add to 180° or more, or in the SSA case the opposite side is shorter than the height b·sin A.
Does it work in radians?
Yes. Switch the angle unit to radians and enter angles such as 1.5708 for 90°. Angles are shown in the chosen unit; the drawing labels use degrees.