Triangle Solver
Fill in any three parts of a triangle and get every remaining side, angle, area, height and centre — drawn to scale, with the rule that was used shown alongside.
What you know
Enter any three parts, at least one of which is a side. Leave the rest blank — they will be filled in.
Sides
Angles (degrees)
Solved triangle
Everything about this triangle
Vertices and centres
Coordinates placing the triangle with vertex A at the origin and side c along the positive x-axis.
| Point | x | y | What it is |
|---|
How to use this calculator
Step by step
- Type any three known parts into the boxes. At least one must be a side — three angles fix the shape but not the size.
- The boxes you filled in are outlined in gold; everything the solver worked out appears in blue. Editing any box clears the solved values and starts again.
- The badge next to the result names the case — SSS, SAS, ASA, AAS or SSA — and the sentence underneath says which rule was applied.
- The drawing is to scale, with vertex A at the origin and side c along the x-axis. Drag to pan and scroll to zoom.
- Tick Show altitudes for the three perpendicular heights, or circumcircle and incircle for the two defining circles.
- Use the preset buttons to load a worked example — including the ambiguous SSA case, which produces two valid triangles.
Good to know
- Angles are in degrees throughout. The three of them always total exactly 180°, and the result panel shows that total so you can check.
- A side is always opposite the angle with the same letter: side a faces angle A. Getting that pairing right is most of the work in setting up a triangle problem.
- The triangle inequality is the quick sanity check on three sides: the longest must be shorter than the other two added together.
- For SSA, do not trust a single answer. If the calculator reports a second triangle, your problem is genuinely ambiguous and needs more information.
- Heron's formula gives the area from three sides alone, with no angle and no height needed — useful for surveying and land measurement.
- The largest angle is always opposite the longest side. If your answer breaks that rule, an input has been mismatched.
The maths behind it
- Law of sines a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R Used for ASA, AAS and SSA.
- Law of cosines c² = a² + b² − 2ab·cos C Used for SAS and SSS; reduces to Pythagoras when C = 90°.
- Heron's formula Area = √( s(s−a)(s−b)(s−c) ), s = (a+b+c) ÷ 2 Area from three sides.
- Area from two sides Area = ½ab·sin C Fastest when you have the included angle.
- Altitude hₐ = 2·Area ÷ a The perpendicular height onto side a.
- Inradius r = Area ÷ s Radius of the circle that just fits inside.
- Circumradius R = abc ÷ (4·Area) Radius of the circle through all three vertices.
- Median mₐ = ½√(2b² + 2c² − a²) From vertex A to the midpoint of side a.
Why does it sometimes tell me there are two triangles?
That is the ambiguous case, SSA: two sides and an angle that is not between them. Geometrically the third side can swing to meet the base in two places, giving one acute and one obtuse triangle, and both satisfy your measurements exactly. The calculator reports both because picking one silently would be guessing. To resolve it you need a fourth piece of information — often knowing whether the triangle is acute or obtuse is enough.
It says my three sides cannot form a triangle.
Then the longest one is at least as long as the other two put together, so the short sides cannot reach across to close the shape. Check for a typo or a unit mismatch — mixing feet with inches is the usual culprit.
Can I use this for a right-angled triangle?
Yes. Enter 90 as one of the angles, or just give the three sides. The general solver handles right triangles without any special mode, and the law of cosines collapses into Pythagoras automatically.
What units are the sides in?
Whatever you like — the calculator never assumes one. Use the same unit for all three sides and every length in the results (heights, medians, radii, perimeter) is in that unit, with the area in square units.
What is the difference between the centroid, circumcentre and incentre?
The centroid is where the medians cross — the point the triangle balances on. The circumcentre is equidistant from all three vertices and is the centre of the circle through them. The incentre is equidistant from all three sides and is the centre of the largest circle that fits inside. They only coincide in an equilateral triangle.