SSS Triangle Calculator

To solve a triangle where three side lengths (a, b, and c) are known, the calculator utilizes a combination of the Law of Cosines and Heron's formula. First, the Law of Cosines is rearranged to find the interior angles. For angle alpha opposite side a, the formula is cos(alpha) = (b^2 + c^2 - a^2) / (2bc). Once alpha and betta are calculated using the inverse cosine function, the third angle gammma is easily found since the sum of angles in a triangle always equals 180 degrees (or pi radians). For the area, Heron's formula is employed: Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter equal to (a+b+c)/2. Finally, the perimeter is simply the arithmetic sum of all three sides (a + b + c).

Let us walk through calculating the properties of a triangle with side lengths a = 5, b = 6, and c = 7. First, we find the perimeter by summing the sides: Perimeter = 5 + 6 + 7 = 18. Next, we find the semi-perimeter s = 18 / 2 = 9. Using Heron's formula, the area is calculated as sqrt(9 * (9-5) * (9-6) * (9-7)) = sqrt(9 * 4 * 3 * 2) = sqrt(216) approximately equal to 14.697 square units. To find angle alpha opposite side a (5), we apply the Law of Cosines: cos(alpha) = (6^2 + 7^2 - 5^2) / (2 * 6 * 7) = (36 + 49 - 25) / 84 = 60 / 84 = 0.7143. Taking the inverse cosine gives alpha approximately equal to 44.42 degrees. Repeating this process for side b yields betta equal to 57.12 degrees, leaving gammma equal to 180 - 44.42 - 57.12 = 78.46 degrees.

Before plugging numbers into any calculator, always check the Triangle Inequality Theorem: the sum of any two side lengths must be strictly greater than the third side. If your sides fail this test (for example, sides of 2, 3, and 6), a valid triangle cannot exist. Additionally, ensure your calculator is set to the correct angular mode (degrees or radians) depending on your application requirements. When drafting designs, double-check your precision settings to avoid compounding rounding errors in multi-step engineering projects.