What is the law of cosines and when is it used?

The law of cosines states that c² = a² + b² - 2ab·cos(C), where a, b, and c are the sides of a triangle and C is the angle opposite side c. It is used when you know all three sides (SSS) or two sides and the included angle (SAS). In the SSS case it lets you find all angles. In the SAS case it finds the unknown side, after which the law of sines resolves the remaining angles. The law of cosines is a generalisation of Pythagoras's theorem: when C = 90°, cos(90°) = 0 and the formula reduces to c² = a² + b².

What is the ambiguous case (SSA) in triangle solving?

The ambiguous case occurs when you know two sides and an angle that is not between those two sides (SSA or the angle-side-side case). Given side a, side b, and angle A (opposite to a), there may be zero, one, or two valid triangles. If a is shorter than the perpendicular height h = b·sin(A), no triangle exists. If a equals h, exactly one right triangle exists. If h is less than a and a is less than b, two different triangles satisfy the given information. If a is greater than or equal to b, exactly one triangle exists. This calculator checks all cases and reports whether a second solution exists.

How do I calculate the area of a triangle from sides and angles?

The most direct formula when you know two sides and the included angle is: Area = (1/2) × a × b × sin(C). For example, with sides of 5 and 7 and an included angle of 60°, the area is 0.5 × 5 × 7 × sin(60°) = 15.16 square units. When you know all three sides you can use Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. Both formulas give the same result once all sides and angles are known.

Triangle Calculator (General)

Solve any triangle from any valid combination of sides and angles. Select how you know your triangle, enter the values you have, and the calculator finds all missing sides, angles, area, and perimeter using the law of cosines and law of sines.

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Standard geometry Law of cosines and law of sines. Handles SSS, SAS, ASA, AAS, and SSA cases including the ambiguous (two-solution) case.

1. Known Information

What do you know?

Side a, angle C (between the sides), side b. Angle C is opposite side c.

Side a (units)

Side b (units)

Side c (units)

Angle A (degrees, opposite side a)

Angle B (degrees, opposite side b)

Angle C (degrees, opposite side c)

2. Triangle Diagram

Side a is opposite angle A, side b opposite B, side c opposite C.

Triangle Solution

Area

Square units

Perimeter
-
Units

Longest Side

Units

Triangle Type

By angles / sides

All Sides

Side a (opposite A)-

Side b (opposite B)-

Side c (opposite C)-

Perimeter (a + b + c)-

All Angles

Angle A (opposite a)-

Angle B (opposite b)-

Angle C (opposite c)-

Sum of angles180.00°

Second Solution (SSA Ambiguous Case)

The given information also describes a second valid triangle:

Side a: -

Side b: -

Side c: -

Angle A: -

Angle B: -

Angle C: -

Area: -

Perimeter: -

Result: Enter your triangle values above.

How to Solve Any Triangle

A triangle has six measurements: three sides (a, b, c) and three angles (A, B, C). You need to know at least three of these (with at least one being a side) to determine the triangle. The two main tools for solving triangles are the law of cosines and the law of sines.

The Law of Cosines

The law of cosines relates all three sides to one of the angles:

c² = a² + b² − 2ab·cos(C)

This is used in two cases:

SAS (Side-Angle-Side): You know two sides and the angle between them. Use the formula to find the third side, then use the law of sines to find the remaining angles.
SSS (Three sides): Rearrange to cos(C) = (a² + b² − c²) / (2ab) to find each angle in turn.

The Law of Sines

The law of sines relates sides to their opposite angles:

a / sin(A) = b / sin(B) = c / sin(C)

This is used in the ASA, AAS, and SSA cases. For ASA and AAS, the third angle is found first (angles sum to 180°), then the remaining sides are found by proportion. SSA requires care because it may produce two solutions.

Worked Example (Default Values)

Given side a = 5, angle C = 60°, side b = 7 (SAS case):

Find side c using the law of cosines: c² = 5² + 7² − 2 × 5 × 7 × cos(60°) = 25 + 49 − 35 = 39, so c = √39 ≈ 6.245
Find angle A using the law of sines: sin(A) = a × sin(C) / c = 5 × sin(60°) / 6.245 ≈ 0.6934, so A ≈ 43.90°
Find angle B = 180° − 60° − 43.90° = 76.10°
Area = ½ × a × b × sin(C) = ½ × 5 × 7 × sin(60°) ≈ 15.155 sq units
Perimeter = 5 + 7 + 6.245 ≈ 18.245 units

These are the values shown in the results when the calculator loads with its default inputs.

The Five Solvable Cases

Case	Known Values	Method	Unique Solution?
SSS	All three sides	Law of cosines for each angle	Yes (if valid triangle)
SAS	Two sides, included angle	Law of cosines, then law of sines	Yes
ASA	Two angles, included side	Third angle = 180 − sum; law of sines	Yes
AAS	Two angles, non-included side	Third angle = 180 − sum; law of sines	Yes
SSA	Two sides, non-included angle	Law of sines; check ambiguous case	0, 1, or 2 solutions

Triangle Types

Once all angles are known, a triangle can be classified:

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

Related Calculators

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Area of a Triangle Calculator: quick area from base and height or other inputs.
Area of a Triangle (Heron's Formula): area from three sides using Heron's formula.
Right Angle Triangle Calculator: solve right-angled triangles using Pythagoras.
SOHCAHTOA Calculator: trigonometry ratios for right-angled triangles.

Method: Law of cosines (c² = a² + b² − 2ab·cos(C)) and law of sines (a/sin(A) = b/sin(B) = c/sin(C)). Area by the formula ½ab·sin(C). SSA ambiguous case checked by comparing the given side against the perpendicular height. Standard Euclidean geometry; angles must sum to 180°.

This calculator works in Euclidean (flat) geometry. Results are to four significant figures. For very small or very large values, verify the result independently. Angle inputs must be in degrees and must sum to less than 180° for any two angles entered.