Why does a triangle have zero diagonals?

In a triangle, every vertex is adjacent to both of the other vertices. Since a diagonal must connect non-adjacent vertices, there are no possible diagonals. The formula confirms this: $\frac{3(3-3)}{2} = 0$.

Diagonals of a Polygon — Definition, Formula & Examples

Diagonals of a polygon are line segments that connect two non-adjacent (non-neighboring) vertices of the polygon. Every polygon with four or more sides has at least one diagonal.

Given a polygon with $n$ vertices, a diagonal is a line segment joining any two vertices that do not share a common side. The total number of distinct diagonals in an $n$ -sided polygon is $\frac{n(n-3)}{2}$ .

Key Formula

D = \frac{n(n - 3)}{2}

Where:

$D$ = Total number of diagonals in the polygon
$n$ = Number of sides (or vertices) of the polygon

How It Works

To find the number of diagonals in any polygon, use the formula

\frac{n(n-3)}{2}

, where

n

is the number of sides. The reasoning behind this formula: each of the

n

vertices can connect to

n - 3

other vertices (you subtract the vertex itself and its two neighbors). This gives

n(n-3)

connections, but since each diagonal is counted twice (once from each endpoint), you divide by 2. A triangle has zero diagonals because every vertex is adjacent to the other two.

Worked Example

Problem: How many diagonals does a hexagon (6-sided polygon) have?

Identify n: A hexagon has 6 sides, so n = 6.

n = 6

Substitute into the formula: Plug n = 6 into the diagonal formula.

D = \frac{6(6 - 3)}{2}

Simplify: Calculate the numerator, then divide by 2.

D = \frac{6 \times 3}{2} = \frac{18}{2} = 9

Answer: A hexagon has 9 diagonals.

Another Example

Problem: How many diagonals does a decagon (10-sided polygon) have?

Identify n: A decagon has 10 sides.

n = 10

Apply the formula: Substitute into the diagonal formula.

D = \frac{10(10 - 3)}{2} = \frac{10 \times 7}{2}

Calculate: Multiply and divide.

D = \frac{70}{2} = 35

Answer: A decagon has 35 diagonals.

Visualization

Why It Matters

Counting diagonals appears frequently in middle school and high school geometry courses, especially when studying polygon properties and interior structures. In more advanced settings, diagonals help triangulate polygons — a technique used in computer graphics to render complex shapes on screen. Understanding this formula also builds your skills in combinatorics, since the counting logic is the same used in many probability problems.

Common Mistakes

Mistake: Using

n(n-1)/2

instead of

n(n-3)/2

Correction: The expression

n(n-1)/2

counts all possible line segments between vertices, including the sides of the polygon. Diagonals exclude the

n

sides, so you subtract 3 (the vertex itself plus its two neighbors) rather than 1.

Mistake: Forgetting to divide by 2

Correction: Each diagonal connects two vertices, so it gets counted once from each end. You must divide by 2 to avoid double-counting.

Related Terms

Diagonal of a Polygon — Defines a single diagonal in detail
Decagon — A 10-sided polygon with 35 diagonals
Dodecagon — A 12-sided polygon with 54 diagonals
Heptagon — A 7-sided polygon with 14 diagonals
Convex — Convex polygons have all diagonals inside
Concave — Concave polygons have some diagonals outside
Apothem — Another interior measurement of regular polygons