How to Find the Number of Diagonals in a Polygon?
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 Maths Explained with Examples
1. What is a diagonal in mathematics?
In mathematics, a diagonal is a line segment connecting two non-adjacent vertices of a polygon or polyhedron. It's a straight line drawn between two corners of a shape that are not already connected by a side.
2. How do you calculate the number of diagonals in a polygon?
The number of diagonals in a polygon with 'n' sides is calculated using the formula: n(n - 3) / 2. For example, a hexagon (6 sides) has 6(6 - 3) / 2 = 9 diagonals.
3. What are the properties of diagonals in different shapes?
Diagonal properties vary depending on the shape:
• In a square and rectangle, diagonals are equal in length and bisect each other.
• In a rhombus, diagonals bisect each other at right angles.
• In a parallelogram, diagonals bisect each other.
• In an isosceles trapezoid, only one diagonal bisects the other diagonal.
4. What is the difference between a diagonal and a side of a polygon?
A side is a line segment forming the perimeter of a polygon. A diagonal connects two non-adjacent vertices (corners) that are not directly next to each other.
5. How do I find the length of a diagonal in a square or rectangle?
Use the Pythagorean theorem:
• For a square with side 's', diagonal length = s√2
• For a rectangle with length 'l' and width 'w', diagonal length = √(l² + w²)
6. Do all polygons have diagonals?
No, triangles do not have diagonals because all vertices are adjacent.
7. Can diagonals be outside a polygon?
Yes, this is possible in concave polygons, where at least one interior angle is greater than 180°.
8. What are some real-world applications of diagonals?
Diagonals are used in construction (bracing structures), engineering (designing stable bridges), and even determining the screen size of TVs (measuring the diagonal distance).
9. How are diagonals used in 3D shapes (polyhedra)?
3D shapes have face diagonals (on faces) and space diagonals (connecting opposite vertices through the interior).
10. What are some common mistakes when calculating diagonals?
Common mistakes include:
• Incorrectly applying the diagonal formula
• Miscounting the number of sides of a polygon
• Failing to distinguish between sides and diagonals
11. What is the difference between the diagonals in a rhombus and a square?
Both have diagonals that bisect each other. However, in a square, the diagonals are equal in length and perpendicular, whereas in a rhombus, the diagonals are only perpendicular and may have unequal lengths.
12. How can diagonals help in finding the area of a shape?
Diagonals are crucial in calculating the area of certain shapes. For instance, the area of a rhombus is (1/2)d1d2, where d1 and d2 are the lengths of the diagonals. Similarly, dividing a parallelogram into two triangles using a diagonal can also aid in area calculation.
