Diagonals of a Polygon
A diagonal of a polygon is a line segment that connects two non-adjacent vertices. In simpler terms, a diagonal joins two corners of a polygon that are NOT next to each other. Sides of a polygon connect adjacent vertices — diagonals connect the remaining ones.
For example, a triangle has 0 diagonals (every vertex is adjacent to every other vertex). A quadrilateral has 2 diagonals. A pentagon has 5 diagonals. As the number of sides increases, the number of diagonals increases rapidly.
In Class 8, you will learn the formula to calculate the number of diagonals in any polygon with n sides. This formula is an important result in the chapter "Understanding Quadrilaterals" and connects to the study of interior and exterior angles of polygons.
What is Diagonals of a Polygon?
Definition: A diagonal of a polygon is a line segment connecting two non-adjacent (non-neighbouring) vertices.
Key observations:
- A side connects two adjacent vertices. A diagonal connects two non-adjacent vertices.
- A polygon with n sides has n vertices.
- From any one vertex, you can draw diagonals to all other vertices except the two adjacent ones and itself.
- So from one vertex, the number of diagonals = n - 3.
Formula:
Number of diagonals = n(n - 3) / 2
Where n = number of sides (or vertices) of the polygon.
Diagonals of a Polygon Formula
Formula for number of diagonals:
D = n(n - 3) / 2
Where:
- D = number of diagonals
- n = number of sides of the polygon
Quick reference table:
- Triangle (n = 3): D = 3(0)/2 = 0
- Quadrilateral (n = 4): D = 4(1)/2 = 2
- Pentagon (n = 5): D = 5(2)/2 = 5
- Hexagon (n = 6): D = 6(3)/2 = 9
- Heptagon (n = 7): D = 7(4)/2 = 14
- Octagon (n = 8): D = 8(5)/2 = 20
- Nonagon (n = 9): D = 9(6)/2 = 27
- Decagon (n = 10): D = 10(7)/2 = 35
Derivation and Proof
Deriving the formula n(n - 3) / 2:
Step 1: A polygon with n sides has n vertices.
Step 2: From any one vertex, you can draw line segments to (n - 1) other vertices.
Step 3: But 2 of these line segments are sides of the polygon (connecting to the two adjacent vertices). These are NOT diagonals.
Step 4: So from one vertex, the number of diagonals = (n - 1) - 2 = n - 3.
Step 5: Since there are n vertices, and each vertex gives (n - 3) diagonals, the total count = n(n - 3).
Step 6: But each diagonal has been counted twice (once from each end). So the actual number of diagonals = n(n - 3) / 2.
Alternative derivation using combinations:
The total number of line segments connecting any two of the n vertices = C(n, 2) = n(n - 1)/2.
Out of these, n segments are the sides of the polygon.
The remaining segments are diagonals = n(n - 1)/2 - n = [n(n - 1) - 2n] / 2 = [n² - n - 2n] / 2 = n(n - 3) / 2.
Types and Properties
Diagonal properties vary across different types of polygons:
1. Triangles (n = 3):
- Number of diagonals = 0.
- A triangle has no diagonals because every pair of vertices is adjacent.
2. Quadrilaterals (n = 4):
- Number of diagonals = 2.
- In a parallelogram, diagonals bisect each other.
- In a rectangle, diagonals are equal.
- In a rhombus, diagonals bisect at right angles.
- In a square, diagonals are equal and bisect at right angles.
3. Regular Polygons:
- In a regular polygon (all sides and angles equal), all diagonals of the same "type" are equal.
- A regular pentagon has 5 diagonals, all of equal length.
- A regular hexagon has 9 diagonals of 2 different lengths — 6 shorter ones and 3 longer ones (the longer ones pass through the centre).
4. Convex vs Concave Polygons:
- In a convex polygon, all diagonals lie entirely inside the polygon.
- In a concave polygon, some diagonals may lie partly outside the polygon.
5. Diagonals from a Single Vertex:
- From one vertex, the number of diagonals = n - 3.
- These diagonals divide the polygon into (n - 2) triangles.
- This is used to derive the angle sum property: sum of interior angles = (n - 2) x 180 degrees.
Solved Examples
Example 1: Example 1: Diagonals of a pentagon
Problem: How many diagonals does a pentagon have?
Solution:
Given: n = 5 (pentagon has 5 sides)
Using the formula:
- D = n(n - 3) / 2
- D = 5(5 - 3) / 2
- D = 5 x 2 / 2
- D = 5
Answer: A pentagon has 5 diagonals.
Example 2: Example 2: Diagonals of an octagon
Problem: Find the number of diagonals in an octagon.
Solution:
Given: n = 8
Using the formula:
- D = 8(8 - 3) / 2
- D = 8 x 5 / 2
- D = 40 / 2
- D = 20
Answer: An octagon has 20 diagonals.
Example 3: Example 3: Diagonals from one vertex of a hexagon
Problem: How many diagonals can be drawn from one vertex of a hexagon?
Solution:
Given: n = 6
From one vertex, number of diagonals = n - 3 = 6 - 3 = 3.
Answer: 3 diagonals can be drawn from one vertex of a hexagon.
Example 4: Example 4: Finding sides from diagonals
Problem: A polygon has 35 diagonals. How many sides does it have?
Solution:
Given: D = 35
Using the formula:
- n(n - 3) / 2 = 35
- n(n - 3) = 70
- n² - 3n - 70 = 0
- (n - 10)(n + 7) = 0
- n = 10 or n = -7
Since n must be positive, n = 10.
Verification: D = 10(7)/2 = 35. Correct.
Answer: The polygon has 10 sides (it is a decagon).
Example 5: Example 5: Triangles formed by diagonals from one vertex
Problem: A polygon is divided into 8 triangles by drawing all diagonals from one vertex. How many sides does the polygon have?
Solution:
Diagonals from one vertex divide the polygon into (n - 2) triangles.
- n - 2 = 8
- n = 10
Answer: The polygon has 10 sides.
Example 6: Example 6: Diagonals of a triangle
Problem: How many diagonals does a triangle have?
Solution:
Given: n = 3
- D = 3(3 - 3) / 2 = 3 x 0 / 2 = 0
This makes sense — in a triangle, every vertex is adjacent to every other vertex, so no diagonals can be drawn.
Answer: A triangle has 0 diagonals.
Example 7: Example 7: Comparing diagonals
Problem: Which has more diagonals — a heptagon (7 sides) or a hexagon (6 sides)? By how many?
Solution:
Heptagon: D = 7(4)/2 = 14
Hexagon: D = 6(3)/2 = 9
Difference = 14 - 9 = 5
Answer: A heptagon has 5 more diagonals than a hexagon.
Example 8: Example 8: Polygon with 54 diagonals
Problem: A polygon has 54 diagonals. Find the number of sides.
Solution:
Given: D = 54
- n(n - 3) / 2 = 54
- n(n - 3) = 108
- n² - 3n - 108 = 0
- (n - 12)(n + 9) = 0
- n = 12 (rejecting n = -9)
Verification: 12 x 9 / 2 = 54. Correct.
Answer: The polygon has 12 sides (a dodecagon).
Example 9: Example 9: Total line segments vs diagonals
Problem: In a hexagon, find the total number of line segments joining any two vertices. How many of these are sides and how many are diagonals?
Solution:
Given: n = 6
Total line segments: C(6, 2) = 6 x 5 / 2 = 15
Sides: 6
Diagonals: 15 - 6 = 9
Verification: D = 6(3)/2 = 9. Correct.
Answer: Total segments = 15, sides = 6, diagonals = 9.
Example 10: Example 10: Diagonals of a 20-sided polygon
Problem: Find the number of diagonals in a polygon with 20 sides.
Solution:
Given: n = 20
- D = 20(20 - 3) / 2
- D = 20 x 17 / 2
- D = 340 / 2
- D = 170
Answer: A 20-sided polygon has 170 diagonals.
Real-World Applications
Diagonal formulas and properties are used in many areas:
- Network Design: In computer networking, if n computers need direct connections, the number of cables needed is C(n,2) = n(n-1)/2. The diagonal formula is a related counting problem.
- Tournament Scheduling: In a round-robin tournament with n teams, each pair plays one match. The total matches = C(n,2). Subtracting the "adjacent" matchups gives a diagonal-like calculation.
- Structural Engineering: Diagonal bracing in polygonal frameworks provides stability. Engineers calculate the number of diagonals needed to make a frame rigid.
- Triangulation: Diagonals from one vertex divide a polygon into triangles. This is used in surveying to calculate the area of irregular polygonal plots.
- Computer Graphics: Polygons are rendered by breaking them into triangles using diagonals. The formula n - 2 gives the minimum number of triangles needed.
- Angle Sum Property: The diagonal formula connects to the angle sum formula — since diagonals from one vertex create (n-2) triangles, the sum of interior angles = (n-2) x 180 degrees.
Key Points to Remember
- A diagonal connects two non-adjacent vertices of a polygon.
- Number of diagonals in an n-sided polygon = n(n - 3) / 2.
- From one vertex, the number of diagonals = n - 3.
- A triangle has 0 diagonals.
- A quadrilateral has 2 diagonals.
- Diagonals from one vertex divide the polygon into (n - 2) triangles.
- Total line segments joining n vertices = n(n-1)/2. Out of these, n are sides and the rest are diagonals.
- In convex polygons, all diagonals lie inside. In concave polygons, some diagonals may go outside.
- The number of diagonals increases rapidly with the number of sides.
- The diagonal formula can be derived using counting principles or combinations.
Practice Problems
- Find the number of diagonals in a nonagon (9 sides).
- A polygon has 44 diagonals. Find the number of sides.
- How many diagonals can be drawn from one vertex of a decagon?
- Find the number of triangles formed by drawing all diagonals from one vertex of an octagon.
- Which polygon has the same number of sides as diagonals?
- A polygon has 90 diagonals. How many sides does it have?
- Find the total number of line segments joining any two vertices of a heptagon. How many are sides and how many are diagonals?
- How many more diagonals does a decagon have than an octagon?
Frequently Asked Questions
Q1. What is a diagonal of a polygon?
A diagonal is a line segment that joins two non-adjacent vertices (corners) of a polygon.
Q2. What is the formula for the number of diagonals?
Number of diagonals = n(n - 3) / 2, where n is the number of sides of the polygon.
Q3. Does a triangle have any diagonals?
No. A triangle has 0 diagonals because every vertex is adjacent to every other vertex.
Q4. How many diagonals does a quadrilateral have?
A quadrilateral has 2 diagonals. Using the formula: 4(4-3)/2 = 4/2 = 2.
Q5. How many diagonals can be drawn from one vertex?
From one vertex of an n-sided polygon, you can draw (n - 3) diagonals.
Q6. Which polygon has 5 diagonals?
A pentagon (5 sides) has 5 diagonals. Using the formula: 5(5-3)/2 = 10/2 = 5.
Q7. Why do we divide by 2 in the formula?
Each diagonal is counted twice — once from each of its endpoints. Dividing by 2 removes this double counting.
Q8. How are diagonals related to the angle sum property?
Diagonals from one vertex divide an n-sided polygon into (n-2) triangles. Since each triangle has an angle sum of 180 degrees, the total interior angle sum = (n-2) x 180 degrees.
Q9. Do all diagonals of a polygon lie inside it?
In a convex polygon, yes — all diagonals lie inside. In a concave polygon, some diagonals may pass outside the polygon.
Q10. Is there a polygon where the number of diagonals equals the number of sides?
Yes. A pentagon has 5 sides and 5 diagonals. It is the only polygon where the number of diagonals equals the number of sides.
