Unit Circle Lab
Law of sines Unit circle

Law of Cosines Calculator

This law of cosines calculator solves SSS and SAS triangles: every side, every angle, the area and the perimeter, step by step, with the triangle drawn to scale.

try

SAS: angle C is between sides a and b · angles in degrees

A 43.9°B 76.1°C 60°a = 5b = 7c = 6.245

Case SAS

c = 6.245 A = 43.9° · B = 76.1° Area 15.155 · perimeter 18.245

Triangle

a
5
b
7
c
6.245
A
43.9°
B
76.1°
C
60°
Area
15.155
Perimeter
18.245

Steps

rounded to 3–4 decimals
  1. You know two sides and the angle between them, so start with the law of cosines for the third side.
  2. c² = a² + b² − 2ab · cos C = 5² + 7² − 2 · 5 · 7 · cos 60° = 39, so c = 6.245.
  3. cos A = (b² + c² − a²) / (2bc) = 0.721, so A = 43.9°.
  4. B = 180° − A − C = 76.1°.
  5. Area = ½ · a · b · sin C = 15.155.

How it works

Two cases, one formula.

  1. 01 · case

    Choose the case

    SAS: two sides and the angle between them. SSS: all three sides.

  2. 02 · solve

    Apply the formula

    SAS gives the third side directly. SSS gives each angle through its cosine, then arccos.

  3. 03 · finish

    Finish the triangle

    The last angle comes from the 180° sum. The panel adds the area and the perimeter.

§1The law of cosines

For any triangle with sides a, b, c and opposite angles A, B, C:

c² = a² + b² − 2ab · cos C

a² = b² + c² − 2bc · cos A · b² = a² + c² − 2ac · cos B

It works for every triangle, acute or obtuse, and it is the tool to use when the law of sines can’t start because no side is paired with its opposite angle.

§2Solving SAS

With two sides and the angle between them, the formula gives the third side right away. Then solve one more angle with the law of cosines and get the last one from the angle sum.

a = 5, b = 7, C = 60°. c² = 25 + 49 − 2 · 5 · 7 · cos 60° = 74 − 35 = 39 → c = √39 ≈ 6.245

cos A = (b² + c² − a²) / (2bc) = (49 + 39 − 25) / (2 · 7 · 6.245) ≈ 0.7206 → A ≈ 43.90°

B = 180° − 43.90° − 60° ≈ 76.10°

§3Solving SSS

With three sides, rearrange the formula for the cosine of each angle, then take arccos:

cos C = (a² + b² − c²) / (2ab)

a = 7, b = 9, c = 12. cos C = (49 + 81 − 144) / 126 = −14/126 ≈ −0.1111 → C ≈ 96.38°

A negative cosine means an obtuse angle, and arccos handles that correctly because its range is 0° to 180°. The inverse cosine calculator shows why. The three sides must also pass the triangle inequality: each side has to be shorter than the other two combined. Sides 1, 2 and 3 fail, since 1 + 2 is not greater than 3, and the calculator says so.

§4Pythagoras as a special case

If C = 90°, then cos C = 0 and the formula becomes c² = a² + b², the Pythagorean theorem. The extra term −2ab · cos C is a correction for angles that are not right. It makes c shorter than the Pythagorean value when C is acute and longer when C is obtuse.

That gives a quick test with three sides: if c² > a² + b² for the longest side c, the triangle is obtuse; if c² = a² + b², it is right; and if c² < a² + b², it is acute.

§5Area with Heron’s formula

The calculator gives the area as ½ · a · b · sin C. With three sides you can also use Heron’s formula: with s = (a + b + c)/2, the area is √(s(s − a)(s − b)(s − c)). For 7, 9 and 12, s = 14 and the area is √(14 · 7 · 5 · 2) = √980 ≈ 31.305, the same value the panel shows.

§6Common mistakes

  • Order of operations: compute 2ab · cos C first and subtract it from a² + b². Don’t multiply (a² + b² − 2ab) by cos C.
  • Pairing the angle with the wrong side: C must be the angle between a and b, opposite c.
  • Forgetting the square root at the end of SAS.
  • Rounding cos C before taking arccos, which can shift the angle by a full degree in flat triangles.

§7FAQ

When should I use the law of cosines instead of the law of sines?

Use it for SSS and SAS, where no side is paired with its opposite angle. After one angle is known, either law can finish the triangle.

Can the law of cosines give two triangles?

No. SSS and SAS always determine a single triangle (or none, if the sides fail the triangle inequality).

Does it work with obtuse angles?

Yes. cos C is negative for obtuse C, so −2ab · cos C becomes positive and c comes out longer, exactly as it should.

What units should I use?

Any length unit, as long as all three sides use the same one. Angles are in degrees.