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Triangle Calculator (SSS, SAS, ASA, AAS, SSA)

Solve any triangle from any three elements. Law of Sines, Law of Cosines, and the ambiguous SSA case.

MATH

Solve any triangle from any three known elements using Law of Sines and Law of Cosines. Supports SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the side between them), AAS (two angles and a non-included side), and the ambiguous SSA case (two sides and a non-included angle - can have 0, 1, or 2 valid solutions).

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Law of Cosines: c² = a² + b² - 2ab·cos(C). SSS and SAS resolve uniquely. ASA and AAS resolve once you find the third angle from A+B+C = 180°. SSA is the ambiguous case - it returns two solutions when sin(B) = b·sin(A)/a allows both an acute B and its supplement 180-B to satisfy the constraints. Output includes all six elements, area (½·a·b·sin(C)), perimeter, and triangle type (equilateral, isosceles, right, acute, obtuse, scalene).

Disclaimer: Assumes Euclidean (flat-plane) geometry. Spherical and hyperbolic triangles use different formulas.

Triangle Calculator (SSS, SAS, ASA, AAS, SSA)

Solve any triangle from any three known elements. Supports SSS, SAS, ASA, AAS, and SSA (ambiguous case) using the Law of Sines and Law of Cosines. Returns all six elements plus area, perimeter, and triangle type.

I know

Side a

Side b

Side c

Sides

a6.000

b8.000

c10.000

Angles

A (opp. a)36.87°

B (opp. b)53.13°

C (opp. c)90.00°

Area24.000

Perimeter24.000

Triangle TypeRight triangle

How Triangle Solving Works

Any triangle has six elements: three sides (a, b, c) and three angles (A, B, C). The angle opposite side a is A, opposite b is B, opposite c is C. To uniquely determine a triangle, you typically need three independent elements - and at least one must be a side (three angles alone only fix the shape, not the size).

Two main laws solve any triangle: Law of Sines a/sin(A) = b/sin(B) = c/sin(C), and Law of Cosines c² = a² + b² - 2ab·cos(C). SSS and SAS use Law of Cosines first to find a third element. ASA and AAS use Law of Sines after finding the third angle (A+B+C = 180°). SSA is the ambiguous case - it can have zero, one, or two valid triangles depending on the input.

The "ambiguous SSA case" occurs when the given angle is acute and the given side opposite that angle is shorter than the height of the would-be triangle. Two configurations can satisfy the constraints. Right and obtuse triangles avoid the ambiguity. When solving SSA by hand, always check both possible solutions and verify which (if any) makes geometric sense.

Results assume Euclidean (flat-plane) geometry. For spherical or hyperbolic triangles, different formulas apply.

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Frequently Asked Questions

What is the difference between SSS, SAS, ASA, AAS, and SSA?

These are the five ways three known elements determine a triangle. SSS = three sides. SAS = two sides and the angle between them. ASA = two angles and the side between them. AAS = two angles and a non-included side. SSA = two sides and a non-included angle (the ambiguous case - can have 0, 1, or 2 valid solutions). Three angles alone (AAA) determines shape but not size, so it cannot uniquely solve a triangle.

When does SSA have two solutions?

When the given angle A is acute AND the side opposite that angle (a) is shorter than the side b but longer than b·sin(A) (the perpendicular height from B). Geometrically, side a can swing to two different positions that both meet the side b. The two solutions are supplementary: B₁ + B₂ = 180°. If a > b, there is exactly one solution. If a < b·sin(A), there is no triangle.

What is the Law of Sines vs the Law of Cosines?

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Used when you know any angle and its opposite side (the "matched pair"). Law of Cosines: c² = a² + b² - 2ab·cos(C). Used when you have all three sides (SSS) or two sides plus the included angle (SAS). Law of Cosines is also the generalization of the Pythagorean Theorem: when C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b².

How do I find the area of a triangle from its sides?

Heron's formula: area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. For a 3-4-5 triangle, s = 6 and area = √(6·3·2·1) = √36 = 6 square units. Alternative when you know two sides and the included angle: area = ½·a·b·sin(C). When you know base and height: area = ½·base·height. All give the same answer; pick whichever fits your known data.

Triangle Calculator (SSS, SAS, ASA, AAS, SSA)

Triangle Calculator (SSS, SAS, ASA, AAS, SSA)

Sides

Angles

How Triangle Solving Works

Frequently Asked Questions

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Sides

Angles

How Triangle Solving Works