Home › Geometry › Triangle Solver

Triangle Solver

Solves a triangle from three sides (SSS) or from two sides and their included angle in degrees (SAS) with the law of cosines, returning all sides, all angles, perimeter, area, inradius, circumradius and the classification by sides and by angles.

When to use

You know three sides, or two sides and the angle between them, and need the remaining sides and angles, the area or the triangle type.

Do not use when: You know two angles and a side (ASA/AAS) or a non-included angle (SSA), which this calculator does not accept; for right triangles from two sides use right-triangle.

Formula

SAS: side_c = √(a² + b² − 2·a·b·cos C). Angles by the law of cosines: cos A = (b² + c² − a²) / (2·b·c), cos B = (a² + c² − b²) / (2·a·c), C = 180° − A − B. area = ½·a·b·sin C; inradius = area / s with s = perimeter / 2; circumradius = a·b·c / (4·area).

Sides must satisfy the triangle inequality. A triangle is classified as right when its largest angle is within 1e-6° of 90°, so rounded side lengths such as 1, 1, 1.4142 classify as acute.

Inputs

Parameter	Type	Unit	Required	Description
`side_a`	number	units	yes	First side. Range: > 0
`side_b`	number	units	yes	Second side. Range: > 0
`side_c`	number	units	no	Third side, opposite the angle between a and b (SSS). Omit for SAS. Range: > 0
`angle_c_deg`	number	degrees	no	Included angle between side_a and side_b in degrees (SAS). Omit when side_c is given. Range: > 0

Outputs

Output	Type	Unit	Description
`side_a`	number	units	Side opposite angle A.
`side_b`	number	units	Side opposite angle B.
`side_c`	number	units	Side opposite angle C (computed for SAS).
`angle_a_deg`	number	degrees	Angle opposite side_a.
`angle_b_deg`	number	degrees	Angle opposite side_b.
`angle_c_deg`	number	degrees	Angle opposite side_c.
`perimeter`	number	units	side_a + side_b + side_c.
`area`	number	units²	½ · side_a · side_b · sin C.
`inradius`	number	units	Radius of the inscribed circle = area / semiperimeter.
`circumradius`	number	units	Radius of the circumscribed circle = a·b·c / (4 · area).
`side_type`	string		equilateral, isosceles or scalene.
`angle_type`	string		acute, right or obtuse.
`method`	string		SSS or SAS.

Example

Sides 3, 4, 5 (SSS): {"side_a":3,"side_b":4,"side_c":5} → {"angle_a_deg":36.8699,"angle_b_deg":53.1301,"angle_c_deg":90,"perimeter":12,"area":6,"inradius":1,"circumradius":2.5,"side_type":"scalene","angle_type":"right","method":"SSS"}

Sides 5 and 7 with 60° between them (SAS): {"side_a":5,"side_b":7,"angle_c_deg":60} → {"side_c":6.245,"angle_a_deg":43.8979,"angle_b_deg":76.1021,"angle_c_deg":60,"perimeter":18.245,"area":15.1554,"inradius":1.6613,"circumradius":3.6056,"side_type":"scalene","angle_type":"acute","method":"SAS"}

Equilateral, side 5: {"side_a":5,"side_b":5,"side_c":5} → {"angle_a_deg":60,"area":10.8253,"inradius":1.4434,"circumradius":2.8868,"side_type":"equilateral","angle_type":"acute"}

GET https://tttkmbb.com/api/v1/calculate/triangle-solver?side_a=3&side_b=4&side_c=5

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/triangle-solver (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/triangle-solver · Markdown: https://tttkmbb.com/geometry/triangle-solver.md · JSON definition: https://tttkmbb.com/geometry/triangle-solver.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="triangle-solver"
OpenAPI operationId: solve_triangle
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Law of cosines (reference)
Wikipedia – Solution of triangles (reference)
Wolfram MathWorld – Law of Cosines (reference)

FAQ

Where is angle C?

Between side_a and side_b, opposite side_c. For SAS enter that included angle; a non-included angle (SSA) can give two solutions and is not supported here.

Why do I get a triangle inequality error?

No triangle exists when one side is as long as or longer than the other two together, e.g. 1, 2, 10. Check the units and the order of the sides.

Related calculators

Triangle Area Calculator — Area only, including the base-and-height method.
Right Triangle Calculator — Simpler solver for right-angled triangles.
Angle Converter — Convert the angles to radians or gradians.