# What is Diagonal — Definition, Formula & Examples in Math

TL;DR

A diagonal is a straight line segment that joins two non-adjacent (non-consecutive) vertices of a polygon or polyhedron. This article gives the formal definition, the count formula n(n−3)/2, length formulas for the square (a√2), rectangle (√(l² + b²)), and cube (a√3), three worked examples (Quick / Standard / Stretch), the properties of diagonals across common quadrilaterals, and the most common mistakes.

A **diagonal** is a line segment that connects two corners of a shape without sliding along any edge. A triangle has none — every pair of vertices is already joined by an edge. A square has two. A pentagon has five.

## The Formal Definition

A **diagonal** of a polygon is a line segment joining two non-adjacent (non-consecutive) vertices. In a polyhedron (a 3D solid), a _space diagonal_ joins two non-adjacent vertices that do not share a face; a _face diagonal_ lies inside a single face of the solid.

The smallest polygon that _can_ have a diagonal is the quadrilateral — three-sided polygons (triangles) have only neighbouring vertices.

> **Quick reference.**
>
> - **Definition:** a segment joining two non-adjacent vertices.
> - **Notation:** d for length; sometimes AC or BD naming the vertices.
> - **Count for an n-sided polygon:** n(n−3)/2.
> - **Length — square (side a):** a√2.
> - **Length — rectangle (l,b):** √(l² + b²).
> - **Length — cube (side a):** a√3 (space diagonal).
> - **Grade introduced:** CCSS-M 4.G.A (polygon properties); NCERT Class 8 — Understanding Quadrilaterals.

## The Count Formula — Why n(n−3)/2

From any one vertex, you can draw a line to every other vertex except two — yourself and your two immediate neighbours (those connect by edges, not diagonals). That gives n−3 diagonals per vertex. With n vertices, the total count of vertex-to-diagonal pairs is n(n−3). But each diagonal has two endpoints, so it's been counted twice — divide by 2.

Number of diagonals=n(n−3)/2.

| Polygon | n | Diagonals |
| --- | --- | --- |
| Triangle | 3 | 0 |
| Quadrilateral | 4 | 2 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 9 |
| Heptagon | 7 | 14 |
| Octagon | 8 | 20 |
| Decagon | 10 | 35 |

The count grows quickly because each new side adds connections to roughly all existing vertices.

## Length of a Diagonal — The Three Most-Used Formulas

These three come up most often in CBSE and Common Core problems.

**Square** (side a). The diagonal cuts the square into two right triangles whose legs are both a. By the Pythagorean theorem:

d = √(a² + a²) = a√2.

**Rectangle** (l,b). The diagonal is the hypotenuse of a right triangle with legs l and b:

d = √(l² + b²).

**Cube** (side a). The _space diagonal_ runs from one corner of the cube straight through to the opposite corner:

d = √(a² + (a√2)²) = a√3.

A pattern emerges: a square's diagonal grows like √2, a cube's like √3, a 4D-hypercube's like 2, and so on. The n√n-growth is one of the cleanest results in elementary geometry.

## Diagonal Properties Across Common Quadrilaterals

The diagonals of a quadrilateral carry rich information about the shape.

| Quadrilateral | Diagonals equal? | Bisect each other? | Bisect at right angles? |
| --- | --- | --- | --- |
| **Square** | ✓ | ✓ | ✓ |
| **Rectangle** | ✓ | ✓ | – |
| **Rhombus** | – | ✓ | ✓ |
| **Parallelogram** | – | ✓ | – |
| **Trapezium (general)** | – | – | – |
| **Isosceles trapezium** | ✓ | – | – |
| **Kite** | – | One bisects the other | ✓ |

These properties are how a geometry student _proves_ what kind of quadrilateral they're looking at — measure the diagonals, check the three boxes above, identify the shape.

## Three Worked Examples — Quick, Standard, Stretch

**Quick.** How many diagonals does a hexagon have?

Use n(n−3)/2 with n=6:

6 × (6−3)/2 = 9.

**Final answer:** 9 diagonals.

**Standard (Wrong Path First — Where Diagonals Trip Students).** A rectangle has length 24 cm and breadth 7 cm. Find the length of its diagonal.

The wrong path: A student tries to add: d=l+b=24+7=31 cm.

The flaw: the diagonal is _not_ the sum of the sides. It's the straight-line distance across, which is the hypotenuse of a right triangle. Adding l and b would only work if the rectangle were collapsed flat.

The rescue: Use the Pythagorean theorem:
d = √(l² + b²) = √(24² + 7²) = 25 cm.

**Final answer:** 25 cm.

**Stretch.** A cube has edge length 6 cm. Find (a) the face diagonal and (b) the space diagonal.

(a) Face diagonal — diagonal of one square face:

d_face = 6√2 ≈ 8.49 cm.

(b) Space diagonal — corner to opposite corner:

d_space = 6√3 ≈ 10.39 cm.

**Final answer:** face diagonal ≈ 8.49 cm; space diagonal ≈ 10.39 cm.

## Conclusion

- A **diagonal** joins two non-adjacent vertices of a polygon or polyhedron.
- An n-sided polygon has n(n−3)/2 diagonals.
- A square's diagonal is a√2; a rectangle's is √(l² + b²); a cube's space diagonal is a√3.
- The diagonal of every rectangle and square is the hypotenuse of a right triangle — Pythagoras decides the length.
- The properties of the two diagonals (equal, bisect, perpendicular) identify the type of quadrilateral.

## Quick Self-Check — Three Problems

1. How many diagonals does a decagon (n=10) have?
2. Find the diagonal of a rectangle of length 99 cm and breadth 12 cm.
3. Find the space diagonal of a cube of side 44 cm.

If you got 21 for problem 1, recount — the formula gives 35.

## Frequently Asked Questions

### What is a diagonal in math?
A line segment joining two non-adjacent vertices of a polygon or polyhedron.

### How many diagonals does an octagon have?
Using n(n−3)/2 with n=8:
8 × 5/2 = 20.

### What is the formula for the diagonal of a square?
d = a√2, where a is the side length.

### Does a triangle have a diagonal?
No. A triangle's three vertices are all adjacent to each other—no non-adjacent pairs exist, so no diagonals can be drawn.

### Do the diagonals of a parallelogram bisect each other?
Yes — that's one of the defining properties. They may not bisect at right angles unless the parallelogram is a rhombus or a square.

### How is the diagonal of a TV screen measured?
From one corner of the screen to the opposite corner, in inches. The advertised size is the diagonal, not the width.

### What's the difference between a face diagonal and a space diagonal?
A face diagonal lies inside one face of a 3D solid; a space diagonal cuts through the interior from one corner to the opposite.