SAS Triangle Calculator — Side-Angle-Side
Enter two side lengths (a and b) and the angle C between them. The calculator uses the Law of Cosines to find the third side c, then solves all three angles and computes the area and perimeter.
°
Third side found by the Law of Cosines: c = √(a² + b² − 2ab·cos C)
- 1
Angle C in radians
60 × π ÷ 180 = 1.047198 - 2
c² = a² + b² − 2ab·cos(C)
7² + 9² − 2 × 7 × 9 × cos(1.047198) = 67Law of Cosines — gives the squared length of the side opposite the included angle. - 3
Side c = √(c²)
√(67) = 8.1854
How does this calculator work?
Given sides a, b and included angle C: find c = √(a² + b² − 2ab·cos C) by the Law of Cosines; recover A = arccos((b² + c² − a²)/2bc); get B = 180° − A − C; compute Area = ½ab·sin C and Perimeter = a + b + c. No ambiguous case — SAS uniquely determines the triangle.
Formula
How this is calculated
The SAS (Side-Angle-Side) configuration is one of the standard triangle congruence cases: knowing two sides and the angle between them uniquely determines the triangle (no ambiguous case). The third side c is found directly by the Law of Cosines: c² = a² + b² − 2ab·cos(C). Unlike the Law of Sines, this formula works even when C is obtuse.
Once c is known, angle A is recovered by applying the Law of Cosines a second time — solving for cos(A) = (b² + c² − a²) / (2bc) and taking the arccosine. This approach is preferred over the Law of Sines for the second angle because arccosine always returns a unique value in [0°, 180°], avoiding the ambiguous case where sin gives two possible angles. Angle B then follows from B = 180° − A − C.
The area is computed directly from the SAS inputs without needing c: Area = ½·a·b·sin(C). The height shown is the altitude from vertex C to the opposite side c, equal to 2·Area / c. All trigonometric functions use exact floating-point arithmetic in radians; results are displayed to four decimal places.
Frequently asked questions
SAS (Side-Angle-Side) means you know two sides of a triangle and the angle that sits between them (the "included" angle). This is sufficient to uniquely determine the triangle — unlike SSA, SAS has no ambiguous case. The third side is found by the Law of Cosines.
Yes. The Law of Cosines handles any angle between 0° and 180°. When C > 90°, cos(C) is negative, so c² = a² + b² − 2ab·cos(C) gives a value larger than a² + b², which is correct — the side opposite an obtuse angle is always the longest side.
The SAS area formula Area = ½·a·b·sin(C) derives from the general definition of area as half the product of two sides times the sine of the included angle. This is exact and requires no intermediate calculation of the third side.
Also known as
TG we-Calculate Editorial Team. (2026). SAS Triangle Calculator — Side-Angle-Side [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sas-triangle-calculator
TG we-Calculate Editorial Team. "SAS Triangle Calculator — Side-Angle-Side." TG we-Calculate. 2026. https://we-calculate.com/calculator/sas-triangle-calculator.
TG we-Calculate Editorial Team, "SAS Triangle Calculator — Side-Angle-Side," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sas-triangle-calculator
@misc{wecalculate_sas_triangle_calculator, title = {SAS Triangle Calculator — Side-Angle-Side}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sas-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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