By Sreela Datta
Updated Aug 30, 2022

In Euclidean geometry, not every trio of segments can form a triangle. The sides must satisfy specific relationships—most notably the triangle inequality theorems, the Pythagorean theorem, and the law of cosines. These principles underpin everything from basic classroom problems to advanced architectural design.

Triangle Inequality Theorem – First Condition

The first theorem states that the sum of any two side lengths must exceed the third. For example, sides of 2 cm, 7 cm, and 12 cm cannot form a triangle because 2 + 7 < 12. Visualize drawing a 12 cm base; the 2 cm and 7 cm segments cannot meet at the other end, confirming the requirement.

Triangle Inequality Theorem – Second Condition

The longest side is always opposite the largest angle. This insight helps identify obtuse, acute, or right triangles: in an obtuse triangle, the side opposite the obtuse angle is the longest. Conversely, the greatest angle lies across from the longest side.

Pythagorean Theorem

For right triangles, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b): c² = a² + b². This timeless result, discovered millennia ago, remains foundational in fields ranging from construction to computer graphics.

Law of Cosines

Generalizing the Pythagorean theorem, the law of cosines applies to all triangles. With sides a, b, c and angle C opposite side c, the relationship is: c² = a² + b² – 2ab·cos C. When C equals 90°, cos C = 0 and the formula reduces to the classic right‑triangle case.

For deeper study, see the Pythagorean theorem and the law of cosines on Wikipedia.