What is the Law of Cosines and when is it used?

The Law of Cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), relating all three sides to one angle. It is used when you know all three sides (SSS) or two sides and the included angle (SAS), because the Law of Sines cannot start the solution in those cases.

What is the Law of Sines and when is it used?

The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C). It is used once at least one angle-side pair is known — typically for ASA, AAS, and as a follow-up step after using the Law of Cosines for SAS.

What is the ambiguous case (SSA)?

When you know two sides and an angle that is NOT between them (SSA), there can be zero, one, or two valid triangles. If sin(B) = b*sin(A)/a is greater than 1, no triangle exists. If it equals exactly 1, there is one right triangle. Otherwise there are two solutions: one with an acute angle B and one with an obtuse angle B. The calculator shows both when they exist.

Do I need to specify units?

No. The calculator is completely unit-agnostic. Enter sides in centimetres, metres, inches, or any other unit — the angles, area (in square units of whatever you chose), and perimeter (in the same linear unit) come out correctly as long as you are consistent.

Why must angles sum to exactly 180 degrees?

This is a fundamental property of Euclidean (flat) geometry. The interior angles of any triangle in a flat plane always add up to exactly 180 degrees. If you enter two angles that together already equal or exceed 180 degrees, no triangle is possible and the calculator tells you so.

Is this calculator accurate for very obtuse or very acute triangles?

Yes, within floating-point precision. The solver clamps acos/asin arguments to the valid range [-1, 1] to prevent NaN from tiny rounding errors, and the Law of Cosines check table lets you verify all three angles independently. For near-degenerate triangles (one angle close to 0° or 180°) the results are still numerically valid, though such triangles have near-zero area.

Triangle Angle Calculator

Every triangle is defined by six measurements — three sides and three angles — but you only need to know three of them (with at least one being a side) to determine the rest completely. This triangle angle calculator does exactly that: pick the combination you know, enter the numbers, and get every missing side, angle, area, and perimeter in one click, with full step-by-step working shown.

How it works

The calculator supports five classic input combinations, each solved by a different formula:

SSS (three sides known) — the Law of Cosines finds each angle in turn:

A = arccos((b^2 + c^2 - a^2) / (2bc))

Repeat the formula cyclically for B and C, then verify the three angles sum to 180°.

SAS (two sides and included angle known) — the Law of Cosines finds the third side:

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

Then the Law of Sines resolves the remaining two angles from the known angle-side pair.

ASA (two angles and the side between them known) — the third angle is simply 180° minus the other two, then the Law of Sines gives both remaining sides:

a = c * sin(A) / sin(C)

AAS (two angles and a non-included side known) — same angle arithmetic as ASA, then the Law of Sines again.

SSA (two sides and a non-included angle) — the ambiguous case. The calculator computes sin(B) = b * sin(A) / a. If this exceeds 1, no solution exists. Otherwise there may be two valid triangles (one with acute B, one with obtuse B); the calculator shows both.

Worked example

Suppose a triangular garden plot has sides of 5 m, 7 m, and 8 m — classic SSS. What are the angles?

Using the Law of Cosines for angle A (opposite the 5 m side):

A = arccos((7^2 + 8^2 - 5^2) / (2 * 7 * 8)) = arccos((49 + 64 - 25) / 112) = arccos(88/112) = arccos(0.7857) = 38.21°

For angle B (opposite 7 m):

B = arccos((5^2 + 8^2 - 7^2) / (2 * 5 * 8)) = arccos((25 + 64 - 49) / 80) = arccos(40/80) = arccos(0.5) = 60°

And angle C = 180° - 38.21° - 60° = 81.79°.

The area by Heron’s formula (or equivalently (1/2)ab*sin(C)) works out to roughly 17.32 m^2 — a triangle you could fence in just 20 m of fencing.

Formula reference

Law	Formula	Used for
Law of Cosines	c^2 = a^2 + b^2 - 2ab*cos(C)	SSS, SAS
Law of Cosines (angle form)	C = arccos((a^2+b^2-c^2)/(2ab))	SSS
Law of Sines	a/sin(A) = b/sin(B) = c/sin(C)	ASA, AAS, SAS follow-up
Angle sum	A + B + C = 180°	All modes
Area (SAS form)	Area = (1/2) * a * b * sin(C)	All modes

The SVG diagram updates live as you type, using your computed (or estimated) angles to scale the shape in real time. The “Law of Cosines check” table at the bottom back-computes all three angles from the final sides so you can confirm the solution is self-consistent.

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