Oblique Triangle Calculator

Solving an oblique triangle relies on two primary geometric frameworks: the Law of Sines and the Law of Cosines. For a triangle with sides a, b, and corresponding opposite angles A, B, and C, the Law of Cosines states that c^2 = a^2 + b^2 - 2ab * cos(C), which is ideal when all three sides (SSS) or two sides and an included angle (SAS) are known. Conversely, the Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C), making it perfect for Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS) scenarios. Additionally, Heron's formula (Area = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter) is used to find the area when all side lengths are available.

Consider solving an oblique triangle using the Side-Angle-Side (SAS) method where side a = 7 cm, side b = 10 cm, and the included angle C = 45 degrees. First, find the third side (c) using the Law of Cosines: c = sqrt(7^2 + 10^2 - 2(7)(10) * cos(45°)). This simplifies to c = sqrt(49 + 100 - 140 * 0.7071) = sqrt(149 - 98.99) = sqrt(50.01), yielding c roughly equal to 7.07 cm. Next, determine the area using the SAS area formula: Area = 0.5 * a * b * sin(C), which gives Area = 0.5 * 7 * 10 * 0.7071 = 24.75 square centimeters.

When working with the Side-Side-Angle (SSA) configuration, always watch out for the ambiguous case where zero, one, or two distinct triangles can be formed. Double-check whether your angle inputs are in degrees or radians before plugging them into trigonometric functions to avoid drastically skewed results. Finally, remember that the sum of all internal angles in any oblique triangle must always equal exactly 180 degrees.