Classifying Triangles Calculator

Triangles are classified into categories using two distinct criteria: side length relationships and interior angle measurements. When side lengths (let us call them a, b, and c) are known, the calculator applies the Law of Cosines to determine interior angles like angle_alpha, angle_beta, and angle_gamma using formulas such as cos(angle_alpha) = (b^2 + c^2 - a^2) / (2 * b * c). Side-based classification looks for equality: an equilateral triangle features three equal sides, an isosceles has two equal sides, and a scalene has no equal sides. Angle-based classification evaluates the maximum interior angle: if all angles are under 90 degrees, it is acute; if one angle equals precisely 90 degrees, it is a right triangle; and if one angle exceeds 90 degrees, it forms an obtuse triangle.

Imagine you have a triangle with side lengths a = 5, b = 7, and c = 10. Because all three side lengths are completely different (5, 7, and 10), we can immediately classify this shape by its sides as a scalene triangle. To find its angle-based classification, we apply the Law of Cosines to find the largest angle opposite to side c (length 10). Plugging the values into the formula: cos(angle_gamma) = (5^2 + 7^2 - 10^2) / (2 * 5 * 7). This simplifies to cos(angle_gamma) = (25 + 49 - 100) / 70, which equals -26 / 70 or approximately -0.3714. Taking the inverse cosine yields an angle_gamma of approximately 111.8 degrees. Because this interior angle is greater than 90 degrees, the calculator successfully classifies this specific shape as an obtuse scalene triangle.

Always verify that your input values satisfy the Triangle Inequality Theorem (the sum of any two side lengths must strictly exceed the length of the remaining side) before attempting any classification. Remember that the sum of all three interior angles in any Euclidean triangle must always equal exactly 180 degrees. If you are calculating missing sides using side-angle-side inputs, double-check your angle mode settings (degrees versus radians) to prevent calculation errors.