Triangle Calculator
Solve any triangle from three sides (SSS), two sides + included angle (SAS), two angles + included side (ASA), two angles + a side (AAS), or two sides + a non-included angle (SSA) — the ambiguous case, handled with both possible solutions shown when they exist. Get every side, angle, the area, perimeter and heights.
Area: 6 · Perimeter: 12
Sides: a=3 · b=4 · c=5 Angles: A=36.87° · B=53.13° · C=90°
Heights: ha=4 (on a) · hb=3 (on b) · hc=2.4 (on c).
Solved via the Law of Cosines and Law of Sines; area via Heron's formula. Units are unit-agnostic (same unit for all sides). How we calculate →
Which triangle case do I have — SSS, SAS, ASA, AAS or SSA?
The case depends on which three pieces of information you know. SSS (three sides), SAS (two sides and the angle between them) and ASA (two angles and the side between them) always give exactly one triangle. AAS (two angles and a side not between them) also gives one triangle. SSA (two sides and an angle that is not between them) is the odd one out — it can produce zero, one or two valid triangles, known as the ambiguous case.
Why does SSA sometimes give two answers?
When you know side a, side b and angle A (opposite a), the Law of Sines gives sin B = b·sin(A)/a. Because sine is positive in both the first and second quadrant, there are usually two angles B (one acute, one obtuse) that satisfy that equation. Both can produce a geometrically valid triangle if the remaining angle C = 180° − A − B is still positive — the calculator checks both and reports whichever are real.
How is the area calculated?
When all three sides are known (directly, or once the case is solved), the calculator uses Heron's formula: with semi-perimeter s = (a+b+c)/2, area = √(s(s−a)(s−b)(s−c)). A 3-4-5 right triangle, for example, has s = 6 and area = √(6×3×2×1) = 6 — matching the simpler ½×base×height check (½×3×4 = 6).
Heights, angles and the full breakdown
Once the three sides and three angles are known, the calculator also derives the perimeter and the three heights (altitudes) — the height dropped onto each side equals 2×area ÷ that side's length. For the 3-4-5 triangle, the height onto side a=3 is 2×6/3 = 4 (which is simply side b, since the right angle sits opposite the hypotenuse).
Frequently asked questions
How do you solve a triangle with three sides (SSS)?
Use the Law of Cosines to find each angle, e.g. cos A = (b²+c²−a²)/(2bc). For sides 3, 4, 5, this gives a right angle (C = 90°) opposite the longest side, and the area from Heron's formula is 6 (perimeter 12).
What is the SSA ambiguous case in trigonometry?
SSA means you know two sides and an angle that is not between them. Because the Law of Sines can yield two valid angles for the unknown angle, there may be 0, 1 or 2 triangles that fit. For a=8, b=10, A=40°, both B≈53.5° and B≈126.5° produce valid triangles, so the calculator returns two solutions.
How do you find the area of a triangle given two sides and an angle (SAS)?
First find the third side with the Law of Cosines, then apply Heron's formula (or use ½·a·b·sin(C) directly). For a=5, b=7 and an included angle of 40°, the third side is about 4.51, and the resulting area is roughly 11.25.
What is Heron's formula?
Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2. It only needs the three side lengths — no angles required — which is why it's the fastest way to get area once an SSS, SAS, ASA, AAS or SSA case has been reduced to three known sides.
Can a triangle have no solution?
Yes, in the SSA case: if the given side opposite the angle is too short to ever reach the other side, no triangle exists. For example, a=3, b=10, A=60° gives sin B = 10×sin(60°)/3 ≈ 2.89, which is impossible (sine can't exceed 1), so there is no valid triangle.
Researched & verified by the Calcuris Data & Research Team. How we build and check our tools →