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About this tool
Solve any triangle from SSS, SAS, ASA, AAS or SSA: all sides, angles, area, perimeter, heights, medians and radii, with a scaled drawing and steps.
The Triangle Calculator solves a triangle from any three known values, as long as one of them is a side. It recognises the case you entered — SSS, SAS, ASA, AAS or SSA — and applies the law of cosines, the law of sines and the 180° angle sum to find the rest. When two sides and a non-included angle allow two different triangles, both solutions are shown. Area uses a numerically stable form of Heron's formula, and heights, medians, angle bisectors, inradius and circumradius are derived from the solved sides. A right-triangle mode fixes angle C at 90°. Everything is calculated in your browser.
Open Triangle Calculator on AltFTool — it loads instantly in your browser.
Choose Any 3 values, or Right triangle to fix angle C at 90° with c as the hypotenuse.
Type three known values into the Sides (a, b, c) and Angles (A, B, C) boxes — for example a = 6, b = 10, A = 30°.
Pick Degrees or Radians and the number of Decimal places; radians also accept input like pi/3.
Read the scaled drawing, the Solution table and the Step-by-step working, switch between Solution 1 and Solution 2 in the SSA case, then Copy results or Download .txt.
Enter any three of the six values. The calculator names the case, solves it, and lists both triangles when the SSA data is ambiguous.
A scaled SVG drawing labels the vertices, side lengths and angles, and marks a right angle with a square, so you can see whether the result looks right.
Each formula used — law of cosines, law of sines, angle sum, Heron's formula, height, median and radius formulas — is listed with your numbers substituted.
Area, perimeter, semiperimeter, all three heights, medians and angle bisectors, inradius, circumradius and the triangle type are given in one place, in degrees and radians.
| Case | You know | Solve with | Number of triangles |
|---|---|---|---|
| SSS | Three sides | Law of cosines for each angle | 1 (if the triangle inequality holds) |
| SAS | Two sides and the angle between them | Law of cosines for the third side | 1 |
| ASA | Two angles and the side between them | Angle sum, then law of sines | 1 |
| AAS | Two angles and a side not between them | Angle sum, then law of sines | 1 |
| SSA | Two sides and an angle not between them | Law of sines for a second angle | 0, 1 or 2 (ambiguous case) |
| AAA | Three angles | Not solvable for size | Infinitely many similar triangles |
Sides are named a, b, c opposite angles A, B, C. Angles must add up to 180°.
First check that the two shorter sides add up to more than the longest one; otherwise no triangle exists. Then use the law of cosines to find each angle: A = arccos((b² + c² − a²) / 2bc), and likewise for B and C. The area follows from Heron's formula using the semiperimeter s = (a + b + c) / 2.
The law of sines says that in any triangle a / sin A = b / sin B = c / sin C, and each ratio equals the diameter of the circumcircle, 2R. It is used when you know an angle and its opposite side, which is the situation in the ASA, AAS and SSA cases.
The law of cosines, c² = a² + b² − 2ab·cos C, links all three sides with one angle. It extends the Pythagorean theorem to any triangle: when C is 90°, cos C is 0 and the formula becomes c² = a² + b². It solves the SSS and SAS cases.
When you know two sides and an angle that is not between them, the side opposite the known angle can swing to meet the base in two places. This happens when that side is shorter than the other known side but longer than the height it must reach. The calculator then lists Solution 1 (acute) and Solution 2 (obtuse).
No. Three angles fix the shape of a triangle but not its size, so infinitely many similar triangles share them. You need at least one side length. If you only know the angles, enter them with any side, such as 1, and the other sides will be in the correct proportion.
With three sides, use Heron's formula: Area = √(s(s − a)(s − b)(s − c)), where s is half the perimeter. With two sides and the angle between them, use Area = ½·a·b·sin C. This calculator solves all sides first and then applies a numerically stable version of Heron's formula.
In ASA the known side lies between the two known angles. In AAS the known side is opposite one of them. Both are solved the same way: the third angle is 180° minus the other two, and the remaining sides come from the law of sines. Both give exactly one triangle.
Look at the largest angle, which is always opposite the longest side. If it is less than 90° the triangle is acute, exactly 90° makes it right, and more than 90° makes it obtuse. With sides only, compare c² with a² + b² for the longest side c.
The inradius r is the radius of the largest circle that fits inside the triangle, touching all three sides; r = Area / s. The circumradius R is the radius of the circle through all three vertices; R = abc / (4·Area). In a right triangle R is exactly half the hypotenuse.
A median joins a vertex to the midpoint of the opposite side. Its length is mₐ = ½·√(2b² + 2c² − a²). The three medians meet at the centroid, which divides each median in the ratio 2:1. A height (altitude) is different: it meets the opposite side at a right angle.
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