Triangle
GeometryA 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). The three inside angles always add up to $180^\circ$.
Example
A yield sign, a slice of pizza, and the shape of a mountain are all triangles. If a triangle has angles of $60^\circ$, $60^\circ$, and $60^\circ$, all three sides are the same length.
Key Insight
The triangle is the simplest polygon and the most rigid shape. If you push on a rectangle at one corner, it collapses, but a triangle keeps its shape - that is why bridges and roofs use triangular supports.
Definition
A triangle is a polygon with three vertices, three sides, and three interior angles summing to $180^\circ$. Triangles are classified by sides (equilateral, isosceles, scalene) or by angles (acute, right, obtuse). For a valid triangle, each side must be shorter than the sum of the other two (triangle inequality).
Example
Triangle with vertices $A(0,0)$, $B(4,0)$, $C(0,3)$: sides $AB = 4$, $BC = 5$, $CA = 3$. It is a right triangle ($3$-$4$-$5$). Area $= (1/2)(4)(3) = 6$. Perimeter $= 12$. Angles: $90^\circ$, approximately $53^\circ$, approximately $37^\circ$.
Key Insight
Triangles are the building blocks of all polygons - any polygon can be divided into triangles (triangulation). This is why triangle area and angle formulas underpin all of polygon geometry.
Definition
A triangle in $\mathbb{R}^2$ is the convex hull of three non-collinear points. Its area via the shoelace formula is $(1/2)|\det([B-A, C-A])|$. The law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ generalizes the Pythagorean theorem. Congruence conditions (SSS, SAS, ASA, AAS) and similarity conditions (AA, SAS, SSS) classify triangles up to isometry and scaling.
Example
By Heron's formula, area $= \sqrt{s(s-a)(s-b)(s-c)}$ where $s = (a+b+c)/2$. For a $3$-$4$-$5$ triangle: $s = 6$, area $= \sqrt{6\cdot3\cdot2\cdot1} = \sqrt{36} = 6$. Confirmed by $(1/2)(3)(4) = 6$.
Key Insight
The triangle inequality (each side less than the sum of the other two) is not just a geometric constraint - it is the defining axiom of a metric space (triangle inequality for distances). Every triangle encodes a metric, and the study of triangles connects Euclidean geometry to metric space theory.