Scalene Triangle
GeometryA scalene triangle has all three sides of different lengths and all three angles of different measures.
Definition
A scalene triangle has all three sides with different lengths and all three angles with different measures. No two sides match and no two angles are equal.
Example
A triangle with sides of $3$ cm, $5$ cm, and $7$ cm is scalene. If you look at it, no side is the same length as another, and no corner is the same angle as another.
Key Insight
The word "scalene" comes from Greek meaning "unequal." Most triangles you draw randomly will be scalene - it is actually the most common type. Equal sides require careful construction.
Definition
A scalene triangle has no two congruent sides and no two congruent angles. The longest side is always opposite the largest angle; the shortest side is always opposite the smallest angle. Scalene triangles have no lines of symmetry.
Example
Triangle with sides $5, 8, 11$ and angles approximately $25^\circ$, $45^\circ$, and $110^\circ$ (obtuse scalene). Since the largest angle ($110^\circ$) is obtuse, the side opposite it ($11$) is the longest, consistent with the side-angle relationship.
Key Insight
The side-opposite-angle relationship (larger angle opposite longer side) holds for all triangles but is most clearly seen in scalene triangles where all three are distinct. This relationship comes from the law of sines: $a/\sin A = b/\sin B = c/\sin C$.
Definition
A scalene triangle is one with pairwise distinct side lengths $a, b, c$ ($a \neq b$, $b \neq c$, $a \neq c$), implying pairwise distinct angles. It has no symmetries (trivial isometry group). The triangle inequality strict: each side strictly less than the sum of the other two. The law of sines $a/\sin A = 2R$ and law of cosines fully determine all elements.
Example
Given sides $a = 6$, $b = 9$, $c = 11$: $s = 13$. Area $= \sqrt{13\cdot7\cdot4\cdot2} = \sqrt{728} = 2\sqrt{182}$ approximately $26.98$. All sides different, so all angles different. Since $11^2 = 121 > 36+81 = 117$, angle $C$ is obtuse.
Key Insight
Scalene triangles are the "generic" case in the moduli space of triangles: the space of triangles up to similarity is a region in $\mathbb{R}^2$ (or equivalently a subset of the upper half-plane), and scalene triangles form an open dense subset. Equilateral and isosceles triangles are the special (measure-zero) boundaries of this space.