Triangle Calculator
Enter the three measurements you know and get the rest of the triangle: every side, every angle, the area and the perimeter. SSA is the one case that can have two answers — both are listed when they exist.
Three sides fix the triangle exactly - there is one answer.
Formula and calculation
Worked example
A triangle with sides 3 m, 4 m and 5 m has an area of 6 square meters and a perimeter of 12 meters. The angle opposite the 3 m side is 36.8699°, and since 3² + 4² = 5² the angle opposite the longest side is exactly 90°.
SSA: the case with two answers
Everything else in triangle solving is polite: three sides fix one triangle, two sides and the angle between them fix one, two angles and a side fix one. SSA does not. If you know sides a and b and angle A — the angle opposite a, not between them — the side a can swing like a gate, and there are often two places it lands. Drop the altitude h = b × sin A from the far vertex and the possibilities line up: shorter than h and side a never reaches, exactly h and it lands square at 90°, and between h and b it hits twice, once before the altitude and once after. The two triangles share sides a and b and angle A, but their angle B values are supplementary — B and 180° − B — which is why the second triangle is wider and shorter. This calculator solves both and lists them in a table instead of quietly returning the first.
Which measurements pin a triangle down
Three of the six values (three sides, three angles) are enough, but not any three. Three angles are not enough — AAA only fixes the shape, not the size, so an infinite family of similar triangles fits and every side length would be a guess. SSA is the only three-value combination that can be ambiguous, and only when the side opposite the angle is shorter than the other known side. If you are solving by hand, work out the largest angle first: the law of cosines is well behaved there, while the law of sines hides the obtuse case inside an arcsin that only returns angles up to 90°.
What the results mean
- Area
- The amount of surface the triangle covers, in square meters, found with Heron's formula from the sides or with ½ × side × side × sin(angle) when two sides and the angle between them are known.
- Perimeter
- The distance all the way around, in meters — the three sides added together.
- Side a
- The side opposite angle A. When two triangles fit the SSA measurements, this row shows the first solution; the table under the result lists both.
- Side b
- The side opposite angle B.
- Side c
- The side opposite angle C. In the ambiguous SSA case this is the side the two solutions disagree about.
- Angle A
- The angle at vertex A, in degrees. A triangle's three angles always add up to exactly 180°.
- Angle B
- The angle at vertex B, in degrees. In the SSA case this is the angle that can take two different values.
- Angle C
- The angle at vertex C, in degrees, worked out as 180° minus the other two.
- Number of solutions
- How many different triangles fit the measurements you entered: normally one, two for the ambiguous SSA case, and none when the numbers cannot describe any triangle at all.
Frequently asked questions
How do I find the area of a triangle with three sides?
Use Heron's formula: add the sides, halve the total to get s, then take the square root of s × (s − a) × (s − b) × (s − c). For sides 3, 4 and 5, s is 6 and the area is √(6 × 3 × 2 × 1) = 6 square meters. The same formula works for any triangle, right-angled or not.
Why does SSA sometimes give two triangles?
Because the known side opposite the known angle can lean two ways. If sides a = 6, b = 10 and angle A = 30°, the altitude from the far vertex is 5, and side a at 6 m is longer than that but shorter than b — so it crosses twice, giving angle B = 56.44° or B = 123.56° and two different triangles.
How do I know if three sides can form a triangle?
Any two sides must add up to more than the third. Sides 3, 4 and 5 work because 3 + 4 > 5, but 1, 2 and 5 do not, since 1 + 2 is only 3. If two sides add up to exactly the third, the triangle flattens to a straight line and its area is zero.
What is the law of cosines used for?
It finds the third side when you know two sides and the angle between them, and it finds an angle when you know all three sides — it is Pythagoras with a correction term, −2bc × cos A, that matters whenever the angle is not 90°. This calculator uses it for the SSS and SAS tabs and for the largest angle in every solution.
Can I solve a triangle with two angles and one side?
Yes. The third angle is 180° minus the other two, and then the law of sines gives the two unknown sides from the one side you know. Two angles and a side is the ASA tab; it always produces exactly one triangle, with no ambiguity.