Diagonals of Polygons

Learn what diagonals are, how to draw them, and discover the formula to count them in any polygon.

Intermediate25 min16 problems
Practice Problems

Definition

A diagonal is a line segment that connects two non-adjacent vertices of a polygon.
Key properties:
  • Diagonals are inside the polygon (for convex polygons)
  • A diagonal connects two vertices that are not next to each other
  • The sides of a polygon are NOT diagonals
Formula for the number of diagonals:
d=n(n−3)2d = \frac{n(n-3)}{2}
where nn is the number of sides (vertices) of the polygon.

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What is a diagonal in a polygon?

Worked Examples

How many diagonals does a quadrilateral (4-sided polygon) have?
1

Identify the number of sides

A quadrilateral has n=4n = 4 sides and 4 vertices → n=4n = 4
2

Apply the formula

d=n(n−3)2=4(4−3)2d = \frac{n(n-3)}{2} = \frac{4(4-3)}{2} → d=4×12d = \frac{4 \times 1}{2}
3

Calculate

d=42=2d = \frac{4}{2} = 2 → d=2d = 2
4

Verify by drawing

Connect A to C, and B to D (the two pairs of opposite vertices) → 2 diagonals confirmed

Common Mistakes

Counting sides as diagonals

Why it's wrong: Sides connect adjacent vertices. Diagonals only connect non-adjacent vertices.

Correct: A diagonal must skip at least one vertex. For a square, AC and BD are diagonals, but AB, BC, CD, DA are sides.

Using n−2n-2 instead of n−3n-3 in the formula

Why it's wrong: From each vertex, you can draw diagonals to all other vertices except itself and its two neighbors.

Correct: The formula is d=n(n−3)2d = \frac{n(n-3)}{2} because each vertex connects to n−3n-3 other vertices (not itself, not the two adjacent).

Forgetting to divide by 2

Why it's wrong: Each diagonal connects two vertices, so counting from both endpoints counts each diagonal twice.

Correct: Always divide by 2: d=n(n−3)2d = \frac{n(n-3)}{2}

Thinking triangles have diagonals

Why it's wrong: In a triangle, every vertex is adjacent to every other vertex.

Correct: Triangles have 0 diagonals: d=3(3−3)2=3×02=0d = \frac{3(3-3)}{2} = \frac{3 \times 0}{2} = 0

Interactive Visual

Polygon explorer

Diagonals from one vertex

6−3=36 - 3 = 3

All diagonals

6(6−3)2=9\dfrac{6(6 - 3)}{2} = 9

From one vertex you reach every vertex except itself and its two neighbours: n − 3 diagonals.

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is a diagonal in a polygon?

Why It Matters

Understanding diagonals helps us:
  • Architecture: Diagonal braces strengthen buildings and bridges
  • Art and Design: Diagonals create dynamic compositions and perspective
  • Engineering: Trusses use diagonal supports for stability
  • Sports: Field markings and court designs use diagonal lines
  • Problem Solving: Dividing polygons into triangles uses diagonals
The diagonal formula is a perfect example of how math can predict patterns!

Real World Applications

Bridge Engineering

Engineers use diagonal cables and supports to strengthen bridges. The pattern of diagonals distributes weight evenly.

Example:

Bridge towers are often stiffened with diagonal cross-bracing. An octagonal frame has 8×52=20\frac{8 \times 5}{2} = 20 diagonals, so a fully cross-braced octagonal tower level could use up to 20 diagonal braces.

Network Connections

In computer networks, the number of direct connections between devices follows a similar pattern to diagonals.

Example:

If 6 computers each need a direct cable to every other one, you need 15 cables: the 6 sides of a hexagon plus its 6×32=9\frac{6 \times 3}{2} = 9 diagonals.

Sports Field Design

Sports pitches and courts are polygons, and their diagonals are used to mark them out accurately.

Example:

A rectangular pitch is a quadrilateral, so it has 4×12=2\frac{4 \times 1}{2} = 2 diagonals. Groundskeepers measure both when marking out a pitch: if the two diagonals are equal, the corners are right angles.

Key Takeaways

  • 1A diagonal connects two non-adjacent vertices of a polygon
  • 2The formula for diagonals is d=n(n−3)2d = \frac{n(n-3)}{2} where nn is the number of sides
  • 3A triangle has 0 diagonals, a quadrilateral has 2, a pentagon has 5, a hexagon has 9
  • 4The number of diagonals grows quickly as the number of sides increases

Frequently Asked Questions

In a triangle, each vertex is adjacent to both other vertices. Since diagonals only connect non-adjacent vertices, there are no vertices to connect with a diagonal. The formula confirms this: 3(3−3)2=0\frac{3(3-3)}{2} = 0.
In a triangle, each vertex is adjacent to both other vertices. Since diagonals only connect non-adjacent vertices, there are no vertices to connect with a diagonal. The formula confirms this: 3(3−3)2=0\frac{3(3-3)}{2} = 0.
Think of it step by step: Each of the nn vertices can connect to n−3n-3 others (not itself, not its 2 neighbors). That gives n(n−3)n(n-3) connections, but we divide by 2 because each diagonal is counted from both ends.
In convex polygons, all diagonals are inside. In concave (non-convex) polygons, some diagonals may pass outside the polygon's interior.

Glossary

Diagonal
A line segment connecting two non-adjacent vertices of a polygon
Vertex
A corner point where two sides of a polygon meet (plural: vertices)
Adjacent vertices
Two vertices that are connected by a side of the polygon
Convex polygon
A polygon where all interior angles are less than 180 degrees and all diagonals lie inside
Today's challengeOne problem, three tries

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