Triangle Side Calculator — Find Missing Side or Angle
Enter two sides and the included angle (SAS) to find the missing third side, or enter all three sides (SSS) to find every angle — powered by the Law of Cosines.
Given information
°
Side opposite to the included angle C, by the Law of Cosines
- 1
a² + b²
5² + 7² = 74 - 2
2ab · cos(C)
2 × 5 × 7 × cos(60°) = 35 - 3
c² = a² + b² − 2ab·cos(C)
74 − 35 = 39 - 4
Side c = √(c²)
√39 = 6.2450
How does this calculator work?
SAS: two sides a, b and included angle C → missing side c = √(a² + b² − 2ab·cos C). SSS: three sides → angle C = arccos((a²+b²−c²)/(2ab)). The Law of Cosines works for any triangle — enter the known values to get every side, angle, area and perimeter instantly.
Formula
How this is calculated
The Law of Cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C), where C is the angle between sides a and b (the included angle) and c is the opposite side. When C = 90° the term −2ab·cos(C) vanishes and the formula reduces to the familiar a² + b² = c².
In SAS mode you supply the two known sides and the angle between them. The calculator evaluates the square root of a² + b² − 2ab·cos(C) to obtain c, then derives the other two angles by re-applying the Law of Cosines. It also computes the area as ½ab·sin(C) and the perimeter.
In SSS mode you supply all three sides and the calculator inverts the formula: cos(C) = (a² + b² − c²) / (2ab), then takes the arc-cosine to find C. The other angles follow by applying the same inversion to the other pairs. The triangle inequality is checked first — if any side equals or exceeds the sum of the other two, no real triangle exists and the result is blank.
Frequently asked questions
The Law of Cosines states that for any triangle with sides a, b, c and angle C opposite to side c: c² = a² + b² − 2ab·cos(C). It generalises the Pythagorean theorem (which applies only when C = 90°) to all triangles.
The included angle in SAS must be the angle sandwiched between the two known sides. If you know two sides and an angle that is NOT between them, use the Law of Sines calculator instead (SSA case), which can have two solutions.
All three angles are computed independently via arc-cosine operations on floating-point numbers, so rounding at the last displayed digit is expected. The true mathematical sum is always exactly 180°.
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Side Calculator — Find Missing Side or Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-side-calculator
TG we-Calculate Editorial Team. "Triangle Side Calculator — Find Missing Side or Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-side-calculator.
TG we-Calculate Editorial Team, "Triangle Side Calculator — Find Missing Side or Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-side-calculator
@misc{wecalculate_triangle_side_calculator, title = {Triangle Side Calculator — Find Missing Side or Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-side-calculator}}, year = {2026}, note = {TG we-Calculate} }
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