The Unit Circle

Law of cosines calculator and triangle solver

Choose what you know and type the numbers. Every side and angle, drawn to scale, with the working; SSA shows both triangles when there are two.

Angles in degrees, sides in any one unit. The working below the drawing substitutes your numbers into each law. Share keeps the case and the values.

Which law the solver uses

The case you choose decides the route[1]:

  • SSS (three sides) and SAS (two sides and the angle between them): the law of cosines, which needs no angle opposite a known side.
  • ASA and AAS (two angles and a side): the third angle is 180° minus the other two, then the law of sines.
  • SSA (two sides and an angle that is not between them): the law of sines, which can give no triangle, one or two. The solver draws every triangle that fits.
c² = a² + b² − 2ab cos C      asin A = bsin B = csin C = 2R(1)

Side a is opposite angle A, and so on; R is the radius of the circle through the three corners. When C = 90°, cos C = 0 and the law of cosines is Pythagoras’s theorem.

Two triangles from one set of numbers

With angle A = 30°, b = 10 and a = 7, the height from the far corner is h = b sin A = 5. Side a is longer than h but shorter than b, so it reaches the base in two places: B ≈ 45.58° or B ≈ 134.42°, and c ≈ 13.56 or 3.76. Both are correct answers to the question as asked. Choose SSA in the solver to see them side by side.

A worked example (SAS)

Sides b = 5 and c = 7 with the angle A = 60° between them. The law of cosines gives a² = 25 + 49 − 2 × 5 × 7 × cos 60° = 74 − 35 = 39, so a = √39 ≈ 6.245. The other angles follow from the law of cosines again: B ≈ 43.90° and C ≈ 76.10°, and the three angles add to 180°.

Inputs, precision and limits

Angles are in degrees, between 0° and 180°, and decimals are kept. Sides must be positive and in the same unit, between a billionth and a billion units; outside that range the box says so and asks you to change the unit. (The angles depend only on the shape, so the solver works on the triangle scaled to a longest side of 1.) If the numbers cannot make a triangle (sides that break the triangle inequality, or two angles that already add to 180° or more), the solver says why instead of drawing something. Results are shown to two or three decimal places. Angles from three sides use Kahan’s numerically stable formula, so even a long thin “needle” triangle gets a positive smallest angle[2]. The triangles are flat; triangles drawn on a sphere follow different laws.

Sources

  1. Abramson, J., et al. (2021). Precalculus 2e (§5.1 Angles, §5.2 Unit circle, §5.3 The other trigonometric functions, §6.1–6.2 Graphs, §6.3 Inverse trigonometric functions, §7.2–7.3 Identities, §8.1–8.2 Non-right triangles, §8.3–8.4 Polar coordinates). OpenStax, Rice University. textbook
  2. Kahan, W. (2014). Miscalculating Area and Angles of a Needle-like Triangle (lecture notes, §1–5 and Table 1). University of California, Berkeley. primary