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.

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.

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.