Triangle Solver – Angle & Side Calculator

Give three triangle calculators the same two sides and one angle, and you might get three different answers. Not because the math is hard — because one specific setup, called SSA, can genuinely produce two correct triangles from the same numbers. At least one popular calculator admits, right in its own help text, that it “may not handle all ambiguous cases correctly.” This one does — and shows you both answers when there really are two.

What Is Triangle Solving?

Every triangle has six measurements: three sides and three angles. Know enough of them, and simple trigonometry fills in the rest. “Solving a triangle” just means finding those missing pieces from what you already have.

The catch is you can’t use any three measurements. Know all three angles and you know the triangle’s shape — but not its size. Triangles of every size share those same three angles. You need at least one side to pin down the actual dimensions.

The Five Ways to Solve a Triangle

Geometry gives every valid triangle problem one of five short names, based on what you’re given and in what order.

SSS — Three Sides

Know all three sides? The Law of Cosines finds every angle directly.

\(\cos(A)=\frac{b^2+c^2-a^2}{2bc}\)

Do this once per angle (or twice — the third angle is just 180° minus the other two).

SAS — Two Sides and the Angle Between Them

Know two sides and the angle squeezed between them? The Law of Cosines finds the third side first, then the Law of Sines fills in the remaining angles.

\(c^2=a^2+b^2-2ab\cos(C)\)

ASA — Two Angles and the Side Between Them

Know two angles and the side connecting them? Subtract from 180° for the third angle, then the Law of Sines finds both remaining sides.

\(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\)

AAS — Two Angles and a Side Not Between Them

Almost identical to ASA — same Law of Sines, just a different starting side. The math doesn’t care which side you started with, only that you have two angles and one side total.

SSA — Two Sides and a Non-Included Angle (The Tricky One)

This is the case that trips up calculators. Two sides and an angle that isn’t between them can describe zero, one, or two different real triangles — the same three numbers, two genuinely different shapes.

Why SSA Can Give Two Answers

Picture side b and angle A fixed in place, with side a swinging like a hinge from the end of angle A. If a is long enough, it can swing down and touch the base in two different spots — two valid triangles, same starting measurements.

Whether that happens depends on comparing a to the height of the swing, h = b × sin(A):

• If a is shorter than h, the swinging side never reaches the base. No triangle at all.
• If a equals h exactly, it touches at exactly one point — a right angle, one triangle.
• If a is longer than h but shorter than b, it crosses the base twice. Two valid triangles.
• If a is at least as long as b, it only crosses once. One triangle.

Worked Example: The Ambiguous Case in Action

Side a = 6, side b = 8, angle A = 30°.

Height check: \(h=8\times\sin(30°)=4\). Since 4 < 6 < 8, this falls in the two-triangle zone.

Solution 1: Angle B ≈ 41.81°, Angle C ≈ 108.19°, side c ≈ 11.40.

Solution 2: Angle B ≈ 138.19°, Angle C ≈ 11.81°, side c ≈ 2.46.

Both are genuinely valid — plug either one back through the Law of Cosines and side a comes out to exactly 6 again, both times. A calculator that only shows one of these isn’t wrong exactly — it’s incomplete, and incomplete is its own kind of wrong when you needed the other answer.

Checking Your Triangle Is Actually Possible

Before trusting any result, two quick checks catch most mistakes:

Triangle inequality — any two sides added together must be longer than the third side. Sides of 3, 4, and 10 can never form a triangle; 3 and 4 don’t reach far enough to close the gap to 10.

Angle sum — every triangle’s three angles add up to exactly 180°, no exceptions. If your known angles already total 180° or more, there’s no room left for a third angle, and no triangle.

Where This Actually Gets Used

Surveyors solve triangles to map land boundaries from a handful of measured points. Engineers use them to lay out roof trusses and structural framing. Navigators use them for triangulation — finding a position from known angles to two landmarks. In every case, the same five setups above cover the whole problem.

Frequently Asked Questions

Bottom Line

Four of the five triangle cases always give one clean answer. The fifth, SSA, sometimes gives two — and a calculator that only shows one isn’t being more certain, it’s just showing less of the picture. Pick the case that matches what you actually know, and trust the number you get.