Triangle
Key Properties
| Property | Statement |
|---|---|
| Area | \(\frac{1}{2}bh\) or \(\frac{1}{2}ab\sin C\) |
| Perimeter | a+b+c |
| Angle sum | 180° |
| Pythagorean | a²+b²=c² (right triangle) |
Examples
Example 1. Find the area of a triangle with base 8 and height 5.
Solution. A = ½·8·5 = 20.
In Depth
Triangles are the simplest polygons and the building blocks of all polygonal geometry. Every polygon can be triangulated — divided into triangles — making triangles fundamental to computational geometry, finite element analysis, and computer graphics.
The law of sines \(a/\sin A=b/\sin B=c/\sin C=2R\) (where \(R\) is the circumradius) and the law of cosines \(c^2=a^2+b^2-2ab\cos C\) solve any triangle given sufficient information. The law of cosines generalizes the Pythagorean theorem to non-right triangles.
Special points: the centroid (intersection of medians) divides each median 2:1 from vertex. The circumcenter (intersection of perpendicular bisectors) is equidistant from all vertices. The incenter (intersection of angle bisectors) is equidistant from all sides. The orthocenter (intersection of altitudes) completes the Euler line with the centroid and circumcenter.
Heron's formula computes area from side lengths alone: \(A=\sqrt{s(s-a)(s-b)(s-c)}\) where \(s=(a+b+c)/2\) is the semi-perimeter. This is useful when the height is unknown.
Further Reading & Context
The study of triangle connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.