Triangle Solver
SSS — sides, angles, area
Enter all three side lengths (SSS)
About This Calculator
The triangle solver finds all unknown sides, angles, and area from any combination of known values using the Laws of Sines and Cosines. It handles all triangle types: acute, right, and obtuse.
Formula
Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Area = (1/2)*a*b*sin(C); Heron: Area = sqrt(s(s-a)(s-b)(s-c)), s=(a+b+c)/2
Example Calculation
Triangle with sides a=7, b=10, c=5
- Find angle C: cos(C) = (7^2+10^2-5^2)/(2*7*10) = (49+100-25)/140 = 124/140 ≈ 0.8857
- C = arccos(0.8857) ≈ 27.66°
- s = (7+10+5)/2 = 11; Area = sqrt(11*4*1*6) = sqrt(264) ≈ 16.25
Angle C ≈ 27.66°; Area ≈ 16.25 sq units
Triangle Classification
| By Angles | Condition | By Sides | Condition |
|---|---|---|---|
| Acute | All angles < 90° | Equilateral | All sides equal |
| Right | One angle = 90° | Isosceles | Two sides equal |
| Obtuse | One angle > 90° | Scalene | No sides equal |
Frequently Asked Questions
What is the minimum information needed to solve a triangle?
You need at least three pieces of information with at least one side: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), or AAS (two angles and any side).
Why can SSA sometimes give two triangles?
The ambiguous case (SSA) occurs when you know two sides and a non-included angle. Depending on the values, there may be 0, 1, or 2 valid triangles that fit those constraints.
What is Heron's formula?
Heron's formula computes a triangle's area from its three side lengths without needing an angle. Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter.