SAS Triangle Calculator

When you are given two sides (let's call them Side A and Side B) and the angle included directly between them (Angle C), you cannot use basic right-triangle trigonometry right away. Instead, the calculator relies on the Law of Cosines to find the third unknown side (Side C): c = √(a² + b² - 2ab · cos(C)). Once the third side is known, the area is computed using the trigonometric formula: Area = ½ab · sin(C). Finally, the remaining unknown angles are derived using the Law of Sines, and the perimeter is simply the sum of all three side lengths (a + b + c).

Imagine you have a triangle where Side A is 4 cm, Side B is 5 cm, and the included Angle C is 30 degrees. First, find Side C using the Law of Cosines: c² = 4² + 5² - (2 · 4 · 5 · cos(30°)). Since cos(30°) ≈ 0.866, this evaluates to c² = 16 + 25 - (40 · 0.866) = 41 - 34.64 = 6.36. Taking the square root gives Side C ≈ 2.52 cm. Next, compute the area: Area = 0.5 · 4 · 5 · sin(30°) = 10 · 0.5 = 5 square centimeters. The perimeter is simply 4 + 5 + 2.52 = 11.52 cm.

Always double-check that your angle is strictly between the two known sides before applying the SAS formulas; if the angle is not the included one, you may be looking at an SSA scenario which can yield ambiguous cases. Ensure your calculator or input mode is set correctly to either degrees or radians depending on your data format. When rounding intermediate steps, keep at least three decimal places to ensure high accuracy for the final perimeter and area calculations.