Triangle
A triangle is a polygon with three sides, three vertices, and three interior angles that always sum to 180 degrees.
Formula
\text{Area} = (1/2) \times \text{base} \times \text{height}; \text{angle sum} = 180^\circ
Definition
A triangle is a flat shape with exactly three straight sides and three corners (vertices), and its three inside angles always add up to $180^\circ$. Triangles are classified by sides (equilateral, isosceles, scalene) or by angles (acute, right, obtuse), and for a valid triangle each side must be shorter than the sum of the other two (the triangle inequality). Formally, a triangle in $\mathbb{R}^2$ is the convex hull of three non-collinear points, congruence conditions (SSS, SAS, ASA, AAS) and similarity conditions (AA, SAS, SSS) classify triangles up to isometry and scaling, and the law of cosines $c^2 = a^2+b^2-2ab\cos C$ generalizes the Pythagorean theorem to any triangle.
Example
A yield sign, a slice of pizza, and the shape of a mountain are all triangles; a triangle with angles $60^\circ$, $60^\circ$, and $60^\circ$ has all three sides equal. For vertices $A(0,0)$, $B(4,0)$, $C(0,3)$: sides $AB=4$, $BC=5$, $CA=3$, a right ($3$-$4$-$5$) triangle with area $6$, perimeter $12$. By Heron's formula, area $=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=(a+b+c)/2$: for this triangle, $s=6$ and area $=\sqrt{6\cdot3\cdot2\cdot1}=\sqrt{36}=6$, confirming the earlier calculation.
Key Insight
The triangle is the simplest polygon and the most rigid shape: push on a rectangle's corner and it collapses, but a triangle keeps its shape, which is why bridges and roofs use triangular supports. Triangles are also the building blocks of all polygons, since any polygon can be divided into triangles (triangulation), and the triangle inequality is not just a geometric constraint, it is the defining axiom of a metric space, so every triangle encodes a metric, connecting Euclidean geometry to metric space theory.