Triangle Equation Solver: Find Sides & Angles

This triangle equation solver takes the values you know about a triangle—side lengths and angles—and calculates the missing sides and angles using the standard laws of trigonometry. You enter three known values (with at least one side), and it returns the remaining sides, angles, and the triangle’s area. The calculator below uses the law of sines, law of cosines, and the angle sum rule to solve any valid triangle you give it.

Formula

The solver relies on three core relationships for any triangle with angles A, B, C and opposite sides a, b, c:

• Angle sum: A + B + C = 180° (or Ï€ radians). This always holds.
• Law of sines: a / sin(A) = b / sin(B) = c / sin(C). Use this when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
• Law of cosines: c² = a² + b² − 2ab·cos(C). Use this when you know two sides and the included angle (SAS) or all three sides (SSS).

In the calculator above, you enter side a, side b, and angle A (the angle opposite side a). The solver first finds angle B using the law of sines: sin(B) = b·sin(A)/a. Then it computes angle C as 180° − A − B, and finally side c using the law of sines again. The area is calculated as ½·b·c·sin(A).

Worked Example

Using the default values: a = 5, b = 6, A = 50°.

1. Find B: sin(B) = b·sin(A)/a = 6·sin(50°)/5 = 6·0.7660/5 = 0.9192. So B = arcsin(0.9192) = 66.8°.
2. Find C: C = 180° − 50° − 66.8° = 63.2°.
3. Find c: c = a·sin(C)/sin(A) = 5·sin(63.2°)/0.7660 = 5·0.8926/0.7660 = 5.83.
4. Area = ½·b·c·sin(A) = 0.5·6·5.83·0.7660 = 13.4 square units.

The calculator gives C = 63.2°, c = 5.83, area = 13.4 (rounded to two decimals).

When to Use a Triangle Solver

You need a triangle solver whenever you have partial information and want the rest. Common real-world cases:

• Construction and carpentry: finding rafter lengths, brace angles, or roof pitches when you know a run and rise.
• Navigation and surveying: determining distances across a river or the height of a building using angle measurements.
• Physics problems: resolving vectors into components or finding resultant forces.
• Geometry homework: checking answers or verifying that a triangle with given sides actually exists.

The solver works for any triangle—right, acute, or obtuse—as long as the given data is consistent and includes at least one side.

How to Choose Which Law to Use

Selecting the correct formula is the most common source of errors. Here is a quick guide:

• You know two angles and one side (AAS or ASA): use the law of sines directly. The third angle is 180° minus the sum.
• You know two sides and the included angle (SAS): use the law of cosines to find the third side, then the law of sines for the other angles.
• You know all three sides (SSS): use the law of cosines to find any angle, then the law of sines for the rest.
• You know two sides and a non-included angle (SSA): this is the ambiguous case—there may be zero, one, or two valid triangles. The law of sines gives a possible angle, but check if the supplementary angle also works.

If your data matches none of these patterns, you do not have enough information to solve the triangle uniquely.

Common Mistakes and Misconceptions

Even with a calculator, errors creep in. The most frequent ones:

• Using degrees instead of radians (or vice versa) in the sine and cosine functions. This solver assumes degrees for angles.
• Mixing up which side is opposite which angle. Side a must be opposite angle A, and so on. Swapping them gives wrong results.
• Forgetting the ambiguous case (SSA). When you know two sides and a non-included angle, there can be two possible triangles. The solver returns one; you must check if the other exists.
• Rounding too early. If you round intermediate angles to whole degrees, the final side can be off by several percent. Keep full precision until the end.
• Assuming a triangle exists. For example, sides 1, 1, and 3 cannot form a triangle because 1 + 1 < 3. The law of cosines would give a negative cosine, which is impossible.

Always verify that your input satisfies the triangle inequality: the sum of any two sides must be greater than the third.

Limitations of the Formula

The laws of sines and cosines are exact for Euclidean geometry, but they have practical limits:

• Rounding errors: For very small angles (near 0°) or very large sides, the law of sines becomes numerically unstable. Using the law of cosines is more robust for side-side-side cases.
• Degenerate triangles: If the given angles sum to exactly 180° but no side is provided, you have a shape but no size—there are infinitely many similar triangles.
• Units: The solver treats sides as plain numbers. If you mix meters and feet, the results are meaningless. Convert all sides to the same unit first.
• Non-Euclidean geometry: On a sphere or curved surface, these formulas do not apply. For large distances on Earth, you would need spherical trigonometry.

For most everyday problems—homework, construction, navigation over short distances—these limitations do not matter.

Alternatives to a Triangle Solver

If you do not have a dedicated solver, you can still work out missing values:

• Right triangles: use the Pythagorean theorem (a² + b² = c²) and the basic trigonometric ratios (sine, cosine, tangent).
• Graphical method: draw the triangle to scale and measure the missing parts with a ruler and protractor. This is accurate to about one degree and one percent.
• Spreadsheet: use built-in functions like SIN, COS, and ASIN in Excel or Google Sheets, but remember to convert degrees to radians (RADIANS function).
• Scientific calculator: many have a “solve triangle” mode, or you can manually apply the laws.

Each method has its own learning curve, but the underlying math is the same.

Frequently Asked Questions

What is the ambiguous case (SSA) and why does it matter?

When you know two sides and an angle that is not between them, the law of sines can give two possible angles because sin(x) = sin(180° − x). For example, if you know a = 5, b = 6, and A = 50°, there is also a valid triangle with B = 113.2° instead of 66.8°. The solver returns one, but you must check whether the other angle also produces a valid triangle (i.e., the remaining angle is positive).

Can I use this solver for right triangles?

Yes. A right triangle is just a special case where one angle is 90°. Enter the known sides and the 90° angle, and the solver will find the rest. However, for right triangles you can often use simpler formulas like the Pythagorean theorem directly.

What if I only know the three angles?

You cannot determine the sides. Any triangle with the same three angles is similar—same shape, but different size. You need at least one side to fix the scale.

Why does the solver sometimes give an error or no solution?

This happens when the input data does not form a valid triangle. Common causes: the angles sum to more than 180°, the triangle inequality is violated (e.g., sides 1, 2, and 4), or the law of sines produces a sine value greater than 1 (which is impossible). Check your inputs for typos or unit inconsistencies.

How do I convert between degrees and radians?

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. This solver expects angles in degrees, so if you have radians, convert them first.

Can I find the area if I only know the three sides?

Yes, use Heron’s formula: s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). This solver does not directly compute Heron’s formula, but you can use the law of cosines to find an angle and then use the area formula ½·ab·sin(C).

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in a | Side a | - | 5 | 0
in b | Side b | - | 6 | 0
in A | Angle A (degrees) | - | 50 | 0
out C | Angle C (degrees) | - | 180 - A - B | 2
out c | Side c | - | (b / sind(A)) * sind(C) | 2
out area | Area | - | 0.5 * b * c * sind(A) | 2