What is the Law of Cosines?

The Law of Cosines relates the lengths of the three sides of a triangle to the cosine of one of its angles. The formula is c² = a² + b² - 2ab·cos(C), where a, b, and c are the side lengths and C is the angle opposite side c. It generalises Pythagoras's theorem: when C is 90 degrees, cos(C) equals zero and the formula reduces to c² = a² + b². You can rearrange the formula to find an unknown angle: cos(C) = (a² + b² - c²) / (2ab).

When should I use the Law of Cosines instead of the Law of Sines?

Use the Law of Cosines when you know two sides and the included angle (SAS configuration), or when you know all three sides (SSS configuration). The Law of Sines is more convenient when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA, though this case can have two solutions). The Law of Cosines avoids the ambiguous case that arises with SSA using the Law of Sines.

How do I find all angles when I know all three sides?

When all three sides are known (SSS), rearrange the Law of Cosines to solve for each angle. For angle A: cos(A) = (b² + c² - a²) / (2bc). For angle B: cos(B) = (a² + c² - b²) / (2ac). For angle C: cos(C) = (a² + b² - c²) / (2ab). Apply the inverse cosine (arccos) to each result to get the angle in degrees. The three angles must sum to exactly 180 degrees. If the cosine value falls outside the range -1 to 1, the three side lengths cannot form a valid triangle.

Cosine Triangle Calculator

Solve any triangle using the Law of Cosines. Choose between SAS (two sides and the included angle) to find the third side and remaining angles, or SSS (all three sides) to find all angles. Results include area, perimeter, and triangle type.

Calculate.co.nz is proud to be partnered with realtor.co.nz, a trusted resource for navigating the New Zealand property market. Their Helpful Articles section offers clear, well-structured insights across buying, selling, and building, making complex real estate topics more accessible. With a focus on up-to-date guidance and practical knowledge, they empower Kiwis to move forward with clarity and confidence in a constantly evolving property landscape.

Standard formula Law of Cosines: c² = a² + b² − 2ab·cos(C). Exact trigonometric identities.

1. Solve Mode

What do you want to solve?

Side a (units)

Side b (units)

Included angle C (degrees, between sides a and b)

Please check your inputs. Ensure all values are positive numbers and the sides can form a valid triangle.

2. Active Formula

c² = a² + b² − 2ab·cos(C)

In SAS mode, sides a and b are the two known sides and C is the angle between them. Side c is the unknown. In SSS mode, all sides are known and the formula is rearranged to find each angle using arccos.

Results

Side c
-
Opposite angle C

Angle A

Opposite side a

Angle B

Opposite side b

Angle C

Opposite side c

Triangle Sides

Side a-

Side b-

Side c-

Perimeter-

Triangle Properties

Angle A-

Angle B-

Angle C-

Area-

Triangle type-

Worked example: With sides a = 5, b = 7 and included angle C = 60°, the Law of Cosines gives c² = 25 + 49 − 2(5)(7)cos(60°) = 74 − 35 = 39, so c = 6.245 units.

The Law of Cosines Explained

The Law of Cosines is a fundamental trigonometric rule that relates the three sides of any triangle to one of its angles. It is most commonly written as:

c² = a² + b² − 2ab·cos(C)

where a, b, and c are side lengths and C is the angle opposite side c. The formula works for any triangle, not just right-angled ones. It is a direct generalisation of Pythagoras's theorem: when C equals 90 degrees, cos(C) = 0 and the formula reduces to c² = a² + b².

You can apply the same formula cyclically to find each angle or side:

a² = b² + c² − 2bc·cos(A)
b² = a² + c² − 2ac·cos(B)
c² = a² + b² − 2ab·cos(C)

To find an angle from three known sides, rearrange to isolate the cosine term:

cos(C) = (a² + b² − c²) / (2ab)

Then apply the inverse cosine function (arccos) to get the angle in degrees.

When to Use the Law of Cosines

Known information	Configuration	Method
Two sides and the included angle	SAS	Law of Cosines: find the third side, then use Law of Sines or Cosines for remaining angles
All three sides	SSS	Law of Cosines rearranged: find each angle using arccos
Two angles and one side	AAS or ASA	Law of Sines is simpler
Two sides and a non-included angle	SSA	Law of Sines (note: may have two solutions)

How the Calculator Works

In SAS mode, you enter sides a and b and the angle C between them. The calculator uses c² = a² + b² − 2ab·cos(C) to find side c, then finds angle A using cos(A) = (b² + c² − a²) / (2bc). Angle B follows from B = 180 − A − C.

In SSS mode, you enter all three sides. The calculator finds angle C = arccos((a² + b² − c²) / (2ab)), then angle A = arccos((b² + c² − a²) / (2bc)), and B = 180 − A − C. The three angles are always verified to sum to 180 degrees.

Area is calculated using the formula Area = (1/2)·a·b·sin(C), which requires only two sides and the included angle.

Triangle Classification

Once all three angles are known, the triangle is classified by its angles and by its sides:

By angles: Acute (all angles under 90°), Right (one angle exactly 90°), or Obtuse (one angle over 90°).
By sides: Equilateral (all three sides equal), Isosceles (two sides equal), or Scalene (no sides equal).

Worked Example

Suppose a = 5, b = 7, and C = 60°. Using the Law of Cosines:

c² = 5² + 7² − 2(5)(7)cos(60°) = 25 + 49 − 70 × 0.5 = 74 − 35 = 39
c = √39 ≈ 6.245 units
cos(A) = (7² + 6.245² − 5²) / (2 × 7 × 6.245) = (49 + 39 − 25) / 87.43 = 63 / 87.43 ≈ 0.7206
A = arccos(0.7206) ≈ 43.9°
B = 180 − 60 − 43.9 = 76.1°
Area = (1/2)(5)(7)sin(60°) = 17.5 × 0.866 ≈ 15.156 square units

Related Calculators

Maths and Stats Calculators
SOHCAHTOA Solver: right-triangle trigonometry using sine, cosine, and tangent ratios.
Triangle Calculator (general): solve triangles using any combination of sides and angles.
Area of a Triangle (Heron's Formula): find triangle area from three side lengths.
Pythagorean Theorem Calculator: right-triangle sides using a² + b² = c².

Sources and method: Law of Cosines: standard trigonometric identity derived from the dot product of vectors. Area formula: (1/2)ab sin(C). Triangle classification follows standard geometric definitions. All calculations use exact floating-point arithmetic via JavaScript Math.cos, Math.acos, and Math.sqrt.

Results are computed to four decimal places. For very large or very small side lengths, floating-point rounding may affect the last digit. Always verify critical engineering calculations independently.