Triangle

A triangle is the simplest polygon — three vertices, three sides, three interior angles. Its angles always sum to 180° (or $\pi$ radians). This single fact powers a vast amount of geometry.

Classification by sides:

• Equilateral: all three sides equal (also all angles $60°$),
• Isosceles: at least two sides equal,
• Scalene: no sides equal.

By angles:

• Acute: all angles $< 90°$,
• Right: one angle $= 90°$,
• Obtuse: one angle $> 90°$.

Right triangles enable the Pythagorean theorem ($a^2 + b^2 = c^2$) and the entire field of trigonometry. The triangle inequality says any side is shorter than the sum of the other two — a fundamental constraint in geometry, vector analysis, and metric spaces.

Area: $A = \frac{1}{2}bh$ (base × height ÷ 2), or Heron's formula when only the three side lengths $a, b, c$ are known: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$.