Triangle Solver

Solve any triangle - find all sides, angles, area, and perimeter from any three known values.

📖 What is a Triangle Solver?

A triangle solver is a calculator that finds all unknown sides and angles of a triangle given any three known values (a combination of sides and angles). There are four main cases, each requiring a different mathematical approach.

Triangles can be solved using two fundamental laws of trigonometry that apply to all triangles, not just right triangles:

Law of Sines relates the sides of a triangle to the sines of their opposite angles. It's ideal when you know two angles and a side (ASA, AAS), or two sides and an opposite angle (SSA).

Law of Cosines is a generalisation of the Pythagorean theorem. It's used when all three sides are known (SSS) or two sides and the included angle are known (SAS).

Together, these two laws can solve any triangle. This calculator handles all four standard cases: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), and AAS (two angles + non-included side).

📐 Formula

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

Law of Sines:

Law of Cosines:

Area:

📖 How to Use This Calculator

Select your known values: SSS (all sides), SAS (two sides + angle between), ASA (two angles + side between), or AAS (two angles + one side).

Enter the known values in the input fields.

Click Solve Triangle - all sides, angles, area, and perimeter are calculated.

💡 Example Calculations

Example 1 - SSS (all sides known)

Sides: a = 5, b = 7, c = 8

Using law of cosines:

A = arccos((49 + 64 - 25)/(2×7×8)) = arccos(88/112) = 38.21°

B = arccos((25 + 64 - 49)/(2×5×8)) = 57.91°

C = 180 - 38.21 - 57.91 = 83.88°

Area = √(10 × 5 × 3 × 2) = 17.32

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Example 2 - ASA (two angles and included side)

A = 40°, side a = 10, C = 70°

B = 180 - 40 - 70 = 70°

Using law of sines: b = 10 × sin(70°)/sin(40°) = 13.47

c = 10 × sin(70°)/sin(40°) = 13.47

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Frequently Asked Questions

What is the law of sines?+

The law of sines states: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides and A, B, C are the opposite angles. It's used when you know two angles and one side (ASA, AAS) or two sides and an angle opposite one of them (SSA).

What is the law of cosines?+

The law of cosines: c² = a² + b² - 2ab·cos(C). It's a generalisation of the Pythagorean theorem for any triangle, not just right triangles. Use it when you know all three sides (SSS) or two sides and the included angle (SAS).

What does SSS, SAS, ASA, AAS mean?+

These abbreviations describe which information you have: S = Side, A = Angle. SSS: all 3 sides known. SAS: 2 sides and the angle between them. ASA: 2 angles and the side between them. AAS: 2 angles and a side not between them.

Can I solve a triangle with two sides and a non-included angle (SSA)?+

Yes, but it's the 'ambiguous case' - there may be 0, 1, or 2 valid triangles depending on the values. This calculator handles it and shows all valid solutions.

How is triangle area calculated from sides and angles?+

Area = (1/2) × a × b × sin(C), where C is the angle between sides a and b. Alternatively, use Heron's formula: Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.

When do you use the law of sines vs law of cosines?+

Use the law of sines when you know: ASA (two angles and one side) or AAS (two angles and one side) or SSA (two sides and one angle - watch for the ambiguous case). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Use the law of cosines when you know: SSS (all three sides) or SAS (two sides and the included angle). Law of cosines: c^2 = a^2 + b^2 - 2ab x cos(C). The law of cosines reduces to the Pythagorean theorem when C = 90 degrees.

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

The ambiguous case occurs in SSA (two sides and an angle opposite one of them) when there may be 0, 1, or 2 valid triangles. Given sides a and b and angle A (opposite side a): if a < b x sin(A), no triangle exists. If a = b x sin(A), exactly one right triangle exists. If b x sin(A) < a < b, two different triangles are possible. If a >= b, exactly one triangle exists. Always check for the ambiguous case when using SSA - this is a common source of errors in trigonometry problems.

How do I find the area of a triangle using only angles and one side (AAS)?+

When you know two angles and one side (AAS or ASA), you can find the area without needing any height. Step 1: find the third angle (180° − A − B). Step 2: use the Law of Sines to find all sides. Step 3: apply Area = (1/2) × a × b × sin(C). Alternatively, Area = a² × sin(B) × sin(C) / (2 × sin(A)), which avoids finding all sides individually. This formula is particularly useful in surveying and navigation where angles are measured directly but distances are harder to obtain.

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

When given two sides and an angle not between them (SSA), there may be 0, 1, or 2 valid triangles. If the side opposite the angle is shorter than the height, no triangle exists. If it equals the height, exactly one right triangle exists. If it is greater than the height but less than the adjacent side, two triangles exist (the ambiguous case). This calculator identifies which case applies and returns all valid solutions.