Diagonals of Polygons
Learn what diagonals are, how to draw them, and discover the formula to count them in any polygon.
Definition
- 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
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Worked Examples
How many diagonals does a quadrilateral (4-sided polygon) have?
Identify the number of sides
A quadrilateral has sides and 4 vertices →
Apply the formula
→
Calculate
→
Verify by drawing
Connect A to C, and B to D (the two pairs of opposite vertices) → 2 diagonals confirmed
Answer: A quadrilateral has exactly 2 diagonals.
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 instead of 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 because each vertex connects to 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:
Thinking triangles have diagonals
Why it's wrong: In a triangle, every vertex is adjacent to every other vertex.
Correct: Triangles have 0 diagonals:
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Practice Problems
16 problemsWhat is a diagonal in a polygon?
Why It Matters
- 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
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 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 diagonals that help define its shape.
Key Takeaways
- 1A diagonal connects two non-adjacent vertices of a polygon
- 2The formula for diagonals is where 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
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