1. What Is a Diagonal?
In a polygon, a diagonal is a line segment that joins two non-adjacent vertices. This means a diagonal connects vertices that are not next to each other along the boundary.
1.1. Basic Idea
If a polygon has many vertices, you can imagine drawing lines between them. But only the lines that connect vertices which are not directly connected by a side are called diagonals.
For example:
- In a triangle, no vertices are non-adjacent → 0 diagonals.
- In a quadrilateral, each vertex can connect to exactly one non-adjacent vertex → 2 diagonals.
2. Diagonals from One Vertex
Take any vertex of a polygon. It cannot connect to itself or its two adjacent vertices to form a diagonal. So diagonals from one vertex are drawn only to the remaining vertices.
2.1. Formula
If a polygon has \(n\) vertices, then from any one vertex you can draw diagonals to:
\(n - 3\)
This is because:
- You cannot draw to the vertex itself.
- You cannot draw to the two neighbours.
2.2. Examples
- Pentagon (\(n=5\)): Diagonals from one vertex = \(5 - 3 = 2\).
- Hexagon (\(n=6\)): Diagonals from one vertex = \(6 - 3 = 3\).
- Octagon (\(n=8\)): Diagonals from one vertex = \(8 - 3 = 5\).
3. Total Number of Diagonals in a Polygon
To find the total number of diagonals in a polygon, we count all the diagonals from each vertex and make sure not to double count.
3.1. Deriving the Formula
Each of the \(n\) vertices gives \((n - 3)\) diagonals. So if we count all at once, we get:
\(n(n - 3)\)
But this counts each diagonal twice (once from each end), so we divide by 2:
\(\displaystyle \text{Total diagonals} = \dfrac{n(n - 3)}{2} \)
3.2. Examples
- Triangle (\(n=3\)): \(\dfrac{3(3-3)}{2} = 0\)
- Quadrilateral (\(n=4\)): \(\dfrac{4(1)}{2} = 2\)
- Pentagon (\(n=5\)): \(\dfrac{5(2)}{2} = 5\)
- Hexagon (\(n=6\)): \(\dfrac{6(3)}{2} = 9\)
4. Diagonals in Common Polygons
Here is a quick reference to help you remember diagonal counts for frequently used polygons:
4.1. Reference Table
| Polygon | Number of Sides (n) | Total Diagonals |
|---|---|---|
| Triangle | 3 | 0 |
| Quadrilateral | 4 | 2 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 9 |
| Heptagon | 7 | 14 |
| Octagon | 8 | 20 |
5. Convex vs Concave Polygons: Diagonal Behavior
Diagonals behave slightly differently in convex and concave polygons.
5.1. In Convex Polygons
All diagonals lie inside the polygon. This makes counting and drawing easier.
5.2. In Concave Polygons
Some diagonals may lie outside the polygon due to its caved-in shape. The counting formulas still work, but diagrams may look different.