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Teacher Guide: Diagonals of Polygons

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

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Polygons. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define what a diagonal is in a polygon
  • Identify and draw all diagonals in polygons up to 8 sides
  • Apply the formula d=n(n−3)2d = \frac{n(n-3)}{2} to calculate the number of diagonals
  • Solve problems involving diagonals, including finding the number of sides given the diagonal count
Prerequisites
  • • Understanding of polygons and their names (triangle, quadrilateral, pentagon, etc.)
  • • Ability to identify vertices and sides of a polygon
  • • Basic multiplication and division skills
Discussion Starters
  • 1. Why do you think the formula uses n−3n-3 instead of n−2n-2 or n−1n-1?
  • 2. If you were building a geodesic dome, why might you want to know about diagonals?
  • 3. Can you think of any everyday objects shaped like polygons where you can see the diagonals?
  • 4. What pattern do you notice in the sequence: 0, 2, 5, 9, 14, 20...? (diagonals for 3, 4, 5, 6, 7, 8 sides)
Common Misconceptions

Every line inside a polygon is a diagonal

Remediation: Emphasize that diagonals MUST connect two vertices. A line from a vertex to a side's midpoint is NOT a diagonal.

Regular and irregular polygons have different numbers of diagonals

Remediation: Show that a regular and irregular hexagon both have 9 diagonals. The shape doesn't matter, only the number of sides.

Differentiation Ideas

For Struggling Students:

  • • Start with physical models using string to connect vertices
  • • Use color-coding: mark vertices one color, diagonals another
  • • Focus on polygons with 4-6 sides before introducing the formula

For On-Level Students:

  • • Derive the formula by counting diagonals from each vertex
  • • Create a table showing sides vs diagonals and find the pattern
  • • Solve word problems involving diagonals in real contexts

For Advanced Students:

  • • Prove why the formula works using combinatorics
  • • Explore how many triangles diagonals from one vertex create
  • • Investigate the intersection points of all diagonals in convex polygons
Standards Alignment
  • 7.G.B.6 (CCSS.MATH.CONTENT.7.G.B.6)

    Solve real-world and mathematical problems involving area, volume, and surface area

  • HSG-CO.A.1 (CCSS.MATH.CONTENT.HSG.CO.A.1)

    Know precise definitions of geometric terms

Lesson Resources
  • visualInteractive Polygon Diagonal Explorer

    Students click vertices to draw diagonals and see the count

  • activityDiagonal Discovery Lab

    Draw polygons from 3 to 8 sides and discover the pattern

  • worksheetDiagonal Calculations

    Practice using the formula with various polygons

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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

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:

A suspension bridge's towers often have diagonal cross-bracing. An octagonal tower section uses 8×52=20\frac{8 \times 5}{2} = 20 potential diagonal supports.

Network Connections

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

Example:

If 6 computers all need direct connections to each other, the number of cables needed is similar to finding connections in a hexagon.

Sports Field Design

Diagonal lines appear in many sports: soccer penalty box corners, baseball diamond, basketball court markings.

Example:

A home plate in baseball is a pentagon, which has 5×22=5\frac{5 \times 2}{2} = 5 diagonals that help define its shape.

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

Why does a triangle have no diagonals?

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.

How do I remember the diagonal formula?

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.

Can diagonals be outside the polygon?

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

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