Triangle Solver Calculator

Solve any triangle from any combination of sides and angles using the law of sines and cosines. Calculates all missing sides, angles, area, perimeter, and triangle type.

The triangle solver calculates all unknown sides and angles of any triangle from the minimum necessary inputs. Given any three elements of a triangle (three angles; two sides and an angle; two angles and a side; or three sides), the calculator applies the law of sines, law of cosines, and angle-sum property to determine all six elements — three sides and three angles. Triangle solving is fundamental to trigonometry, navigation, surveying, architecture, and engineering: GPS triangulation, roof truss design, map reading, and construction setout all reduce to triangle-solving problems. The area is also computed from the solved triangle using Heron's formula (from three sides) or the ½ab sin C formula (from two sides and an included angle).

Select your known information type: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), AAS (two angles and non-included side), or SSA (two sides and non-included angle — may have 0, 1, or 2 solutions).
Enter the known values — sides in any unit (cm, m, inches), angles in degrees or radians.
The calculator solves for all remaining sides and angles and displays the full triangle with area and perimeter.
For the SSA (ambiguous) case: the calculator flags whether 0, 1, or 2 valid triangles exist and displays all valid solutions.
Check the result visually: all angles must sum to 180°; the longest side must oppose the largest angle.

Triangle solving formulas

Angle sum: A + B + C = 180°

Law of Sines: a ÷ sin A = b ÷ sin B = c ÷ sin C

Law of Cosines: c² = a² + b² − 2ab cos C (and cyclic permutations)

Area (Heron's): s = (a+b+c)÷2; Area = √(s(s−a)(s−b)(s−c))

Area (SAS): Area = ½ × a × b × sin C

Right triangle special case: Pythagorean theorem c² = a² + b²; sin A = a/c, cos A = b/c, tan A = a/b.

Example SAS: a = 7, b = 10, C = 40°. c² = 49 + 100 − 2×7×10×cos40° = 149 − 140×0.766 = 149 − 107.2 = 41.8 → c = 6.47. Area = ½×7×10×sin40° = 35×0.643 = 22.5 sq units.

Reading your triangle solution

Triangle types and properties

Triangle classification by angles: acute (all angles below 90°); right (one angle exactly 90° — Pythagorean theorem applies); obtuse (one angle above 90°). By sides: equilateral (all sides equal, all angles 60°); isosceles (two sides and two angles equal); scalene (all sides and angles different). The SSA case is called the "ambiguous case" because it can produce 0, 1, or 2 valid triangles: if the given side opposite the given angle is shorter than the height (b × sin A), no triangle exists; if exactly equal to the height, exactly one (right) triangle exists; if longer than the height but shorter than the adjacent side, two triangles exist. The calculator handles all cases automatically.

Math tips and best practices

For right triangles, use the simpler SOH-CAH-TOA trigonometric ratios before reaching for the law of cosines — it's faster and less error-prone.
When using the law of cosines to find an angle (given three sides), always use the rearranged form: cos C = (a² + b² − c²) ÷ (2ab) — never the solving form first.
Always verify that the solved angles sum to exactly 180° (within rounding tolerance) as a sanity check.
In the ambiguous (SSA) case, draw a rough sketch before calculating — it makes it much easier to see whether 0, 1, or 2 triangles should be expected.
For surveying and navigation, always work in consistent angle units (all degrees or all radians) — mixing them is a common error source.

The largest triangle in practical use is geodetic — the Earth's surface triangle formed by three GPS satellites in orbit, each ~20,000 km above Earth, with sides of roughly 50,000 km.
An equilateral triangle has the maximum area for a given perimeter — it is the most area-efficient triangular shape.
The 3-4-5 right triangle (sides in ratio 3:4:5) was used by ancient Egyptian rope-stretchers (harpedonaptai) to set right angles in construction over 4,000 years ago.
Heron of Alexandria (c. 10–70 AD) published the formula for triangle area from three sides — it remains the most elegant and widely used method 2,000 years later.

Common mistakes to avoid

Using the law of sines to solve for an obtuse angle — sin θ = sin(180° − θ), so the law of sines cannot distinguish between an acute and obtuse angle; use the law of cosines when the angle type is uncertain.
Forgetting that the SSA case can yield two valid triangles — always check for both solutions before assuming uniqueness.
Rounding intermediate calculations — keep full precision through all steps and only round the final answer to avoid cascading errors, especially in obtuse triangles where cosines can be negative and small errors compound.
Confusing degrees and radians in calculator input — ensure your scientific calculator is in the correct angle mode (DEG vs RAD) before computing trigonometric functions.

Triangle solver results are mathematical computations based on the inputs provided. For professional surveying, structural engineering, navigation, and construction applications, always verify calculations independently and apply appropriate safety factors. Surveying measurements are subject to instrument accuracy and environmental conditions; professional surveys must be conducted by licensed surveyors in accordance with national standards. This calculator is for educational and planning purposes and does not constitute professional engineering or surveying advice.

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 side lengths and A, B, C are the opposite angles.

What is the law of cosines?

The law of cosines states c² = a² + b² − 2ab·cos(C). It generalises the Pythagorean theorem for any triangle.

What does SSS, SAS, ASA mean?

These are triangle congruence cases: SSS = three sides known; SAS = two sides and included angle; ASA = two angles and included side; AAS = two angles and non-included side.

How do I find the area of a triangle without the height?

Use Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2, or use Area = ½ab·sin(C) if two sides and an angle are known.

What is an obtuse triangle?

An obtuse triangle has one angle greater than 90°. The law of cosines must be used to solve it, since the law of sines can give an ambiguous result.