Triangle Solver | Law of Sines & Cosines

Solve any triangle given SSS, SAS, ASA, or AAS using law of sines and cosines.

Known Information

Three sides known

Side a

Side b

Side c

What Is the Triangle Solver | Law of Sines & Cosines?

A triangle has six elements, three sides and three angles. Given any three that include at least one side, the remaining three can be determined. Different known combinations require different solution methods: the Law of Sines (angle-side relationships) or the Law of Cosines (side-side-angle / side-side-side).

Input Combinations (Cases)

›SSS (3 sides): use Law of Cosines to find all angles
›SAS (2 sides + included angle): Law of Cosines for third side, then Law of Sines
›ASA or AAS (2 angles + a side): third angle = 180° − A − B, then Law of Sines
›SSA (2 sides + non-included angle): ambiguous case, may have 0, 1, or 2 solutions

The SSA Ambiguous Case

›Given sides a, b and angle A (not the included angle), two triangles may exist
›If sin B = b·sin(A)/a < 1, compute B₁ = arcsin(sinB) and B₂ = 180° − B₁
›B₂ is valid only if C₂ = 180° − A − B₂ > 0 (remaining angle must be positive)
›The calculator detects and displays both solutions when the ambiguous case arises

Formula

Law of Cosines

c² = a² + b² − 2ab·cos(C)

Law of Sines

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

Area (SAS)

Area = ½ab·sin(C)

Heron's Formula

s=(a+b+c)/2; Area=√(s(s−a)(s−b)(s−c))

Inradius

r = Area / s

Circumradius

R = abc / (4·Area)

How to Use

1Select the input case: SSS, SAS, ASA, AAS, or SSA.
2Enter the required values (sides in any consistent unit, angles in degrees).
3Click Solve, all remaining sides, angles, area, and properties are computed.
4For SSA: if two solutions exist, both triangles are displayed with an ambiguous-case note.
5The SVG diagram updates to show your triangle with labelled sides and angles.
6Computed properties include: area, perimeter, inradius, circumradius, altitudes, and type classification.

Example Calculation

SAS case: side a = 7, side b = 10, included angle C = 40°:

Step 1, Law of Cosines for side c: c² = 7² + 10² − 2(7)(10)·cos(40°) = 149 − 140 × 0.7660 = 41.76 c = 6.462 Step 2, Law of Sines for angle A: sin A = 7 × sin(40°) / 6.462 = 0.6957 A = 44.05° Step 3, Angle B: B = 180° − 40° − 44.05° = 95.95° Area = ½ × 7 × 10 × sin(40°) = 22.50

Result

Sides: 7, 10, 6.46, Angles: 44.1°, 96.0°, 40°. Area = 22.50 square units. Classification: Obtuse Scalene (one angle > 90°, all sides different).

Understanding Triangle | Law of Sines & Cosines

Triangle Solving, Input Cases Quick Reference

Case	Given	Method	Unique?	Notes
SSS	3 sides	Law of Cosines	Yes (if valid)	Check triangle inequality first
SAS	2 sides + included angle	Law of Cosines → Sines	Yes	Most common case in practice
ASA	2 angles + included side	Law of Sines	Yes	Third angle = 180° − A − B
AAS	2 angles + non-incl. side	Law of Sines	Yes	Same as ASA once third angle found
SSA	2 sides + non-incl. angle	Law of Sines	0, 1, or 2	The ambiguous case
AAA	3 angles only	Not uniquely solvable	No	Infinite similar triangles

Frequently Asked Questions

What is the SSA ambiguous case in triangle solving?

SSA (two sides + non-included angle) is the only case that can be ambiguous:

›If sin B = b·sin(A)/a > 1: no triangle exists (the arc cannot reach the opposite side)
›If sin B = 1: exactly one right triangle exists
›If sin B < 1: two possible triangles, B₁ = arcsin(sinB) and B₂ = 180° − B₁
›B₂ gives a second valid triangle only if C₂ = 180° − A − B₂ > 0

What is the Law of Cosines used for?

›SSS: rearrange to find angle, cos C = (a² + b² − c²) / (2ab)
›SAS: find the third side, c² = a² + b² − 2ab·cos(C)
›When C = 90°: cos(90°) = 0 → c² = a² + b², reduces to Pythagorean theorem
›More numerically stable than the Law of Sines for obtuse angles

What is the Law of Sines used for?

›Relates each side to the sine of its opposite angle: a/sin A = b/sin B = c/sin C
›ASA: two angles known → third angle = 180° − A − B; then use ratio for remaining sides
›AAS: same approach, find third angle, then apply the sines ratio
›SSA: use to find angle B from sin B = b·sin(A)/a, then check for the ambiguous case

What is the inradius of a triangle?

›The incircle is the largest circle that fits entirely inside the triangle
›Formula: r = Area / s, where s = (a + b + c) / 2 (semi-perimeter)
›Every triangle has exactly one incircle, its centre is the triangle's incentre
›For an equilateral triangle with side a: r = a / (2√3) ≈ 0.289a

What is the circumradius?

›The circumcircle passes through all three of the triangle's vertices
›Formula: R = abc / (4·Area) = a / (2·sin A)
›For a right triangle: R = c/2, where c is the hypotenuse (Thales' theorem)
›The circumcentre is equidistant from all three vertices, it's the centre of the circumcircle

How are triangle types classified?

›By angles, Acute: all angles < 90°; Right: one angle = 90°; Obtuse: one angle > 90°
›By sides, Equilateral: all three sides equal (also equiangular, all 60°)
›By sides, Isosceles: exactly two sides equal (base angles are also equal)
›By sides, Scalene: all three sides different (all angles also different)

What is the triangle inequality?

›Rule: each side must be strictly less than the sum of the other two
›Equivalently: the longest side must be less than the sum of the two shorter sides
›Violation example: sides 3, 4, 10 fail because 10 > 3 + 4 = 7, no triangle possible
›This is validated automatically for SSS input, the calculator will show an error if violated

Related Calculators

Right Triangle Solver | Sides & Angles

Solve any right triangle given two sides or one side and one angle.

Area Calculator | 11 Shapes with Formulas & Perimeter

Calculate area and perimeter for 11 shapes: rectangle, square, circle, triangle, right triangle, trapezoid, parallelogram, rhombus, ellipse, hexagon, and sector. Formulas shown with your values substituted in.

Trigonometry Calculator | Sin, Cos, Tan

Calculate sine, cosine, tangent, and their inverses for any angle.