Triangle Calculator
Solve every side, angle, perimeter, and area of a triangle from SSS, SAS, ASA, AAS, or SSA (including the ambiguous SSA case).
Details
Area
6
in square units
This finds every missing side and angle of a triangle from the values you know, along with the area and perimeter.
It handles any valid combination — three sides, two sides and an angle, two angles and a side — and tells you when the information given does not pin down a single triangle.
How a triangle is solved
A triangle has six measurements: three sides and three angles. Knowing the right three of them fixes the other three completely.
Which three matters enormously. Three sides define exactly one triangle. Three angles define a shape but not a size, so they define infinitely many.
The angles always add to 180°, so knowing two gives you the third for free. That is why 'two angles and a side' is enough to solve everything.
The angle has to sit between the two known sides for this to work. Give the angle somewhere else and the triangle may not be uniquely determined at all.
What to enter
- Given
- Which combination you know: three sides, two sides and the angle between, two angles and a side, and so on.
- Sides a, b, c
- Each side is labelled with the lowercase letter matching the uppercase angle opposite it. Side a is opposite angle A.
- Angles A, B, C
- In degrees. They always total 180°, so the calculator fills in the third automatically.
Which combinations actually solve a triangle
- SSS — three sides
- Always gives exactly one triangle. Angles come from the law of cosines.
- SAS — two sides and the angle between
- Always one triangle. Law of cosines finds the third side.
- ASA / AAS — two angles and any side
- Always one triangle, since the third angle is fixed. Law of sines does the rest.
- SSA — two sides and an angle NOT between
- The ambiguous case. It can give two triangles, one, or none.
- AAA — three angles
- Fixes the shape but not the size. Infinitely many similar triangles fit.
What this assumes
Angles are in degrees.
A valid triangle needs any two sides to add to more than the third. 2, 3 and 9 cannot form one.
How to calculate a triangle's sides and angles
Two rules cover every case. Use the law of sines when you have a matched side and angle pair, and the law of cosines when you do not.
- law of cosines
- For three sides, or two sides and the angle between them
- law of sines
- When you have a side and its opposite angle
Identify what you have. Label sides with lowercase and angles with uppercase, keeping each side opposite its matching angle.
Pick the right rule. A matched side-and-angle pair means the law of sines. Otherwise, the law of cosines.
Use the angle sum. Once you have two angles, the third is 180° minus their total.
Check it is sensible. The longest side must sit opposite the largest angle. If it does not, something has gone wrong.
See a worked example: two sides of 5 and 6 with a 60° angle between them
- Side a
- 5
- Side b
- 6
- Angle C
- 60°, between them
Law of cosines: c² = 5² + 6² − 2 × 5 × 6 × cos(60°).
cos(60°) = 0.5, so c² = 25 + 36 − 30 = 31.
c = √31 = 5.5678.
Area is easier here: ½ × a × b × sin(C) = ½ × 5 × 6 × 0.8660 = 12.99.
Third side 5.57, area 12.99
Frequently asked questions
Three measurements, but not any three. Three sides, two sides and the angle between them, or two angles and any side each give exactly one triangle.
Three angles are not enough: they fix the shape but not the size, so infinitely many triangles fit.
It happens with SSA: two sides and an angle that is not between them. That information can describe two different triangles, one, or none at all.
Picture swinging the known side from a fixed point — it may cross the opposite line in two places. This is where exam answers get lost, because students find one solution and stop.
Use the law of sines when you have a side and the angle directly opposite it. Use the law of cosines when you do not, which means three sides, or two sides with the angle between.
A practical tip: start with the law of cosines to find the largest angle first. That avoids the ambiguity the law of sines can introduce with obtuse angles.
With two sides and the angle between them: area = ½ × a × b × sin(C).
With all three sides, use Heron's formula. On a 3-4-5 triangle it gives exactly 6, matching ½ × 3 × 4 since that one happens to be right-angled.
No. Any two sides must add to more than the third, which is called the triangle inequality.
Sides of 2, 3 and 9 cannot close: the short two laid end to end still fall short of the long one. The calculator will tell you when a set is impossible.
Problems people actually run into
Stopping at the first answer in the ambiguous case
With two sides and a non-included angle, the law of sines can produce two valid triangles, because sin(30°) and sin(150°) are identical.
Always check whether the supplementary angle also gives a workable triangle. If the angles still total under 180°, it is a second genuine solution and it usually carries marks of its own.
Mismatching the side and angle labels
Side a must be opposite angle A. Labelling them the wrong way round breaks both the law of sines and the law of cosines, and produces answers that look plausible.
The check that catches it: the largest angle must be opposite the longest side. If your answer contradicts that, the labels are crossed.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: September 4, 2026