Diagonals of Polygons Formula Properties and Solved Examples
The concept of diagonals in maths plays a key role in mathematics and geometry, helping students to solve problems related to polygons, quadrilaterals, and 3D shapes in both textbooks and real life.
What Is Diagonals in Maths?
A diagonal in maths is a straight line segment that connects any two non-adjacent (not next to each other) vertices of a polygon. In simple words, diagonals are "slanting lines" that go across the shape instead of sticking to the edges. You’ll find this concept used while counting lines inside polygons, calculating areas and sides in shapes like parallelograms, squares, rectangles, and in identifying geometric patterns.
Key Formula for Diagonals in Maths
Here’s the standard formula to find the number of diagonals in any polygon:
\( \text{Number of diagonals} = \frac{n(n-3)}{2} \)
where n is the number of sides (or vertices) in the polygon.
Types of Diagonals
| Type | Example | Description |
|---|---|---|
| Interior Diagonal | Diagonal in a convex polygon | Lies fully inside the polygon (e.g., square, pentagon) |
| Exterior Diagonal | Concave polygon | May partially go outside the shape |
| Major Diagonal | Rectangle | Longest diagonal in the shape |
| Minor Diagonal | Rhombus | Shortest diagonal (when the shape has two lengths) |
Diagonals in Common Shapes
| Shape | Number of Diagonals | Formula/Property |
|---|---|---|
| Triangle | 0 | No non-adjacent vertices |
| Rectangle | 2 | Both are equal length |
| Square | 2 | Equal and perpendicular |
| Pentagon | 5 | Use formula with n=5 |
| Hexagon | 9 | Use formula with n=6 |
| Decagon | 35 | n=10, formula |
Step-by-Step Illustration
Let’s see how to use the diagonal formula to solve a typical problem:
1. **Question:** How many diagonals does a regular octagon have?2. First, count the number of sides: An octagon has n = 8.
3. Apply the formula: Number of diagonals = n(n-3)/2
4. Substitute values: 8 × (8−3) / 2 = 8 × 5 / 2 = 40 / 2 = 20
5. **Final Answer:** There are 20 diagonals in a regular octagon.
Key Properties of Diagonals in Maths
- The diagonal always joins non-adjacent corners.
- In a regular polygon, all diagonals are equal in length.
- Diagonals may be equal, perpendicular, or bisect each other depending on the shape (e.g., in squares and rhombuses).
- Parallelogram diagonals bisect each other but are not necessarily equal.
- In a rhombus, diagonals bisect at 90° and are not equal.
Cross-Disciplinary Usage
Diagonals in maths are not only useful for geometry but also help in Physics (vectors, forces in frames), Computer Science (graphics, mesh designs), and even in building structures. For exams like JEE, Olympiad, and NEET, understanding diagonals is vital for questions on geometry, area, and logical reasoning.
Speed Trick or Vedic Shortcut
Shortcut for Diagonals in a Polygon: Just plug in 'n' in the formula n(n-3)/2. For remembering, "Each vertex connects to all others except itself and two neighbors."
- Multiply the number of sides by (number of sides - 3).
- Divide by 2 for the final count.
Tricks like this make fast work of MCQ questions. Vedantu’s live classes show more exam-solving hacks and tips.
Try These Yourself
- How many diagonals are there in a 12-sided polygon?
- Find the diagonal length of a square with side 7 cm.
- Count the diagonals in a pentagon by drawing.
- Which quadrilateral’s diagonals are equal and bisect at right angles?
Frequent Errors and Misunderstandings
- Counting sides as diagonals (only non-adjacent vertices count!)
- Using the formula wrongly: mixing up n(n-1)/2 with n(n-3)/2.
- Forgetting triangles have 0 diagonals.
- Confusing diagonal length formula for square (use: \( d = a\sqrt{2} \)), not n(n-3)/2.
Relation to Other Concepts
The idea of diagonals in maths connects closely with polygons, types of quadrilaterals, and concepts like polygon angles and area of triangles. Understanding diagonals also helps in learning about diagonal matrices and symmetry.
Classroom Tip
An easy way to remember the diagonal formula: "Take the number of sides, subtract 3, multiply, then divide by 2". Teachers at Vedantu often use colored markers to connect corners on real drawings—try this for better understanding!
We explored diagonals in maths: from the basic definition and formulas to properties, types, common errors, and cross-subject uses. Keep practicing and use helpful tips from Vedantu’s teachers to get confident at counting and drawing diagonals—essential for clear learning and fast problem-solving in geometry!
Read more: Square | Rectangle | Rhombus | Types of Quadrilaterals | Diagonal Matrix
FAQs on Diagonals in Geometry Meaning and Applications
1. What is a diagonal in geometry?
A diagonal is a line segment that joins two non-adjacent vertices of a polygon. In simple terms, it connects corners that are not next to each other.
- In a square, a diagonal connects opposite corners.
- In a pentagon, diagonals connect vertices that skip one vertex in between.
- Diagonals are drawn inside the polygon and are different from its sides.
2. What is the formula for the number of diagonals in a polygon?
The formula for the number of diagonals in an n-sided polygon is n(n − 3) / 2.
- Here, n represents the number of sides (or vertices).
- Subtract 3 because each vertex connects to itself and two adjacent vertices, which do not form diagonals.
- Divide by 2 to avoid double counting.
3. How many diagonals does a quadrilateral have?
A quadrilateral has 2 diagonals. Using the formula n(n − 3)/2 with n = 4:
- 4(4 − 3)/2 = 4 × 1 / 2
- = 2 diagonals
4. What is the length of the diagonal of a square?
The length of the diagonal of a square is a√2, where a is the side length. This is derived using the Pythagoras theorem.
- For a square with side length a, the diagonal forms a right triangle.
- Diagonal² = a² + a² = 2a²
- Diagonal = √(2a²) = a√2
5. How do you find the diagonal of a rectangle?
The diagonal of a rectangle is found using the formula √(l² + w²), where l is length and w is width. This comes from the Pythagorean theorem.
- Square the length and width.
- Add them together.
- Take the square root.
6. Do the diagonals of a square bisect each other?
Yes, the diagonals of a square bisect each other at right angles. This means:
- Each diagonal cuts the other into two equal parts.
- They intersect at 90°.
- They are equal in length.
7. What are the properties of diagonals of a rectangle?
The diagonals of a rectangle are equal in length and bisect each other, but they do not intersect at right angles. Key properties include:
- Both diagonals are the same length.
- They cut each other into two equal parts.
- The angles formed at the intersection are not necessarily 90° (unless it is a square).
8. What are the properties of diagonals of a rhombus?
The diagonals of a rhombus bisect each other at right angles and are not equal in general. Important properties are:
- They intersect at 90°.
- Each diagonal divides the other into two equal parts.
- They bisect the interior angles of the rhombus.
9. How do you find the number of diagonals in a pentagon?
A pentagon has 5 diagonals, calculated using the formula n(n − 3)/2.
- Here, n = 5.
- 5(5 − 3)/2 = 5 × 2 / 2
- = 5 diagonals
10. What is the difference between a side and a diagonal of a polygon?
A side connects two adjacent vertices, while a diagonal connects two non-adjacent vertices of a polygon.
- Sides form the boundary of the polygon.
- Diagonals lie inside the polygon.
- Every polygon has as many sides as vertices, but the number of diagonals depends on n(n − 3)/2.
