Area
6u²
Area enclosed by the shape.
Mathematics · Geometry
Triangle solver — sides, angles and area
Solve a triangle from SSS, SAS, ASA or AAS data. Find all sides and angles, area and perimeter; detect impossible triangles.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Perimeter
12u
Total boundary length.
Side a
3u
Side opposite angle A.
Side b
4u
Side opposite angle B.
Side c
5u
Side opposite angle C.
Angle A
36.87°
Opposite side a.
Angle B
53.13°
Opposite side b.
Angle C
90°
Opposite side c.
Results show up to four significant digits for readability; internal calculations retain full precision.
See calculation details
01Angle A
Formula
A = arccos((b² + c² − a²) / (2bc))Substitution
arccos((4² + 5² − 3²) / (2 × 4 × 5)) × 180°/πResult
36.87
02Angle B
Formula
B = arccos((a² + c² − b²) / (2ac))Substitution
arccos((3² + 5² − 4²) / (2 × 3 × 5)) × 180°/πResult
53.13
03Angle C
Formula
C = 180° − A − BSubstitution
180° − 36.87° − 53.13°Result
90
04Perimeter
Formula
P = a + b + cSubstitution
3 + 4 + 5Result
12
05p
Formula
p = P/2Substitution
12 / 2Result
6
06Area
Formula
S = √(p(p − a)(p − b)(p − c))Substitution
√(6 × (6 − 3) × (6 − 4) × (6 − 5))Result
6
01Formulas and symbols
Formulas used
a
a/sin A = b/sin B = c/sin C
A + B + C = 180°
a, b, c: sides opposite angles A, B, C (in degrees).
c
c² = a² + b² − 2ab cos C
C: included angle between a and b.
S
S = √(p(p−a)(p−b)(p−c))
p = (a+b+c)/2
S: area; p: semiperimeter.
| Symbol | Meaning |
|---|---|
| a | Side a (u) |
| b | Side b (u) |
| c | Side c (u) |
| A | Angle A (°) |
| B | Angle B (°) |
| C | Angle C (°) |
| S | Area |
| P | Perimeter |
02Validation example
Reference numerical case
- SSS a = 3, b = 4, c = 5 gives perimeter 12 u, area 6 u² and C = 90°. An equilateral triangle of side 2 u has area √3 u² and angles of 60°.
03How it works
Sides a, b, c are opposite angles A, B, C. SAS uses a, b and included angle C. ASA uses A, B and c; AAS uses A, B and a. The remaining data follow from the cosine or sine law and the angle sum.
The cosine law relates the two known sides and their included angle C to opposite side c; in SSS it also recovers each angle.
Heron’s formula uses the semiperimeter p and the three sides to compute the area without requiring a height.
04Limits and conventions
Positive sides in one common unit; angles in degrees. SSS must satisfy strict triangle inequalities. SSA is excluded because it can have two solutions. Degenerate triangles and angles rounded to zero are rejected.
05Common mistakes
Use the included angle C in SAS, not an arbitrary angle. A side and its opposite angle must keep the same letter.
06Practical questions
How many measurements are needed to solve a triangle?
Usually three independent measurements are needed, with at least one side known. Valid combinations include SSS, SAS, ASA and AAS; three angles alone determine shape but not scale.
Why can some side-and-angle combinations produce two triangles?
The ambiguous SSA case can correspond to two different triangles because the sine law may admit two supplementary angles with the same sine. This calculator excludes SSA rather than silently choosing one solution.
07Technical references
- Precalculus 2e — non-right triangles and trigonometry — OpenStax
- Precalculus 2e — law of sines — OpenStax