Triangle Sum Theorem Calculator

The Triangle Sum Theorem states that the interior angles of any flat (Euclidean) triangle always add up to exactly 180 degrees (or π radians). Mathematically, if you have three interior angles denoted as alpha (α), beta (β), and gamma (γ), their relationship is expressed as α + β + γ = 180°. To find a missing angle, simply subtract the sum of the two known angles from 180°. For instance, if you are solving for gamma, the formula is γ = 180° - (α + β).

Imagine you are designing a roof truss and need to cut a wooden beam. You measure two of the interior triangle angles as 40° and 75° (let us call these angle alpha and angle beta). To find the third missing angle (gamma), apply the triangle sum theorem formula. First, add the two known angles together: 40° + 75° = 115°. Next, subtract that sum from the total interior sum of 180°. Calculating 180° - 115° gives you 65°. Therefore, the third angle is precisely 65°, ensuring your structural layout is accurate.

Always double-check your angle unit settings before calculating, ensuring you are working consistently in either degrees or radians. Remember that this theorem strictly applies to Euclidean geometry; non-Euclidean triangles on curved surfaces will have angle sums greater or less than 180°. If your calculated third angle results in a negative number or zero, your input angles are invalid for a real triangle.