Classifying Polygons and Triangles
A polygon is a closed figure made entirely of straight line segments. No curves allowed, and the shape must fully close. Each point where two sides meet is called a vertex. A triangle — the simplest polygon — has 3 sides and 3 vertices. You can't build a polygon with fewer than 3 sides, which is why triangles are where everything starts.
Classifying Polygons by Side Count
Polygons are named by the number of sides they have:
| Sides | Name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 8 | Octagon |
| 10 | Decagon |
| 12 | Dodecagon |
Polygons with more than 12 sides are typically called n-gons — a 56-sided polygon would just be called a 56-gon.
Regular, Equilateral, and Equiangular
These three terms describe special properties a polygon can have:
- An equilateral polygon has all sides of equal length. An equilateral triangle is the most familiar example.
- An equiangular polygon has all angles of equal measure. A rectangle is equiangular (all four angles are 90°), but its sides don't have to be equal.
- A regular polygon is both equilateral and equiangular. A square is a regular polygon. A regular hexagon has six equal sides and six equal angles.
Classifying Triangles by Their Sides
Triangles get their own naming system based on side lengths:
- Scalene — all three sides have different lengths
- Isosceles — two sides are equal (and the angles opposite those sides are also equal)
- Equilateral — all three sides are equal (and all angles are 60°)
Classifying Triangles by Their Angles
You can also classify a triangle by its largest angle:
- Acute — all three angles are less than 90°
- Right — one angle is exactly 90°
- Obtuse — one angle is greater than 90°
These two classification systems are independent — a triangle can be both isosceles and obtuse, for example, or scalene and acute.
Medians and Altitudes
Two more terms that come up frequently when working with triangles:
A median is a segment drawn from one vertex to the midpoint of the opposite side. Every triangle has three medians, one from each vertex.
An altitude (or height) is a segment drawn from one vertex perpendicular to the opposite side.
In a right triangle, two of the sides are actually altitudes. The altitude is what you use when calculating the area of a triangle — area = ½ × base × height, where the height is the altitude to that base.
The distance from a point to a line is always measured along the perpendicular — that's the shortest possible path, and it's what the altitude represents.