Diagonals

Introduction to Diagonals

In geometry, diagonals are important parts of a shape. A diagonal is a straight line that connects two non-adjacent vertices of a polygon or a solid figure. In simple words, there is a slanting line that goes from one corner of a shape to another corner that is not directly next to it. Diagonals help us study the structure and properties of different shapes in mathematics.

The word "diagonal" comes from the Greek word "diagonios", which means "from angle to angle". Diagonals can be seen in many polygons, such as squares, rectangles, rhombuses, and parallelograms. They also have applications in algebra; for example, the diagonal square matrix represents the elements that connect a corner to the opposite corner. In this article, we will explore the meaning of diagonals, their properties, formulas, and how they appear in different shapes with clear examples.

Table of Contents

• What are Diagonals?
• Diagonals Formula
• Diagonals of Shapes
• Number of Diagonals
• Length of Diagonals
• Solved Problems
• FAQs on Diagonals

What are Diagonals?

A diagonal is a straight line that connects two corners (vertices) of a shape, but not ones already joined by a side. It is always a straight line that goes across the shape, but not up, down, or along the sides.

In simple words, a diagonal joins two opposite corners of a polygon (like a square, rectangle, or pentagon) or a solid shape (like a cube). 

Diagonals Formula

If a polygon has "n" vertices (corners), the number of diagonals can be found using the formula:

Number of Diagonals = n(n−3)2

Example: Let us find the number of diagonals in a pentagon (5 sides).

• Number of Diagonals =  5(5−3)2

• Number of Diagonals =  5×22

• Number of Diagonals =  5

• So, a pentagon has 5 diagonals.

Diagonals of Shapes

In geometry, different shapes have different numbers of diagonals. A diagonal is a line that connects two corners (vertices) that are not next to each other. Let's study the diagonals in various spaces.

1. Diagonals of Triangle

• A triangle has 3 sides and 3 corners. Since every corner is next to another, there are no non-adjacent vertices. 

• So, a triangle does not have any diagonals.

• Number of diagonals in a triangle = 0

2. Diagonals of Square

• A square has 4 equal sides and 4 corners. A diagonal of a square connects one corner to the opposite corner.

• A square has 2 diagonals. Both diagonals are of the same length. They cross each other at the center and divide the square into equal parts. 

• Number of diagonals in a square = 2

• Formula to find the diagonal of a square:

• Diagonal = a2

• Where “a” = side of the square.

3. Diagonals of Rectangle

• A rectangle has 4 sides and 4 corners. Just like a square, a rectangle has 2 diagonals. They are equal in length and cut each other into 2 equal parts at the center.

• Number of diagonals in a rectangle = 2

• Formula to find the diagonals of a rectangle:

• Diagonal = l2+b2

• Where “I” = length, “b” = breadth 

4. Diagonals of Rhombus

• A rhombus has 4 equal sides. It has 2 diagonals. The diagonals of a rhombus cut each other at right angles (90°). They also divide the rhombus into 4 right-angled triangles.

• Number of diagonals in a rhombus = 2

• If the area of the rhombus is A, and the diagonals are p and q, then Area = 12×p×q

• From this, diagonals can be found as: p=2×Areaq, q=2×Areap

5. Diagonals of Parallelogram 

• A parallelogram has 4 sides, where opposite sides are equal. It has 2 diagonals. They cut each other into equal halves, but they are not always of the same length.

• Number of diagonals in a parallelogram = 2

• Formulas for diagonals:

  • d1=2a2+2b2−d22

  • d2=2a2+2b2−d12

6. Diagonals of Pentagon

• A pentagon has 5 sides and 5 corners. A pentagon has 5 diagonals that join the non-adjacent corners.

• Number of diagonals in a pentagon = 5

7. Diagonals of Hexagon

• A hexagon has 6 sides and 6 corners. A hexagon has 9 diagonals when we join the non-adjacent corners.

• Number of diagonals in a hexagon = 9

8. Diagonals of Cube

A cube is a 3D shape with 6 square faces, 12 edges, and 8 corners. A cube has two types of diagonals:

• Face diagonals: Lines that connect opposite corners of each square face. (12 in total)

• Space diagonals: Lines that connect opposite corners passing through the inside of the cube. (4 in total)

• Total diagonals of cube = 12 + 4 = 16

9. Diagonals of Cuboid

• A cuboid is also a 3D shape with 6 rectangular faces, 12 edges, and 8 corners. Just like a cube, a cuboid also has two types of diagonals:

• Face diagonals: 12 in total

• Space diagonals: 4 in total

• Total diagonals of cuboid = 12 + 4 = 16

Number of Diagonals

Length of Diagonals

Solved Problems

Example 1: Diagonal of a square 

Problem: Find a class diagonal that is 6 cm.

Solution:

• Side square, a = 6 cm

• Formula: Diagonal = a2

• Substitute d=62

• 2≈1.414→d=6×1.414=8.484cm

• Diagonal = 8.484 cm

Example 2: Diagonal of a rectangle 

Problem: Find a diagonal of a rectangle whose length is 8 cm and breadth is 6 cm.

Solution:

• Length, L = 8 cm; Breadth, b = 6 cm

• Formula: diagonal = l2+b2

• Substitute : d=82+62=64+36=100

• Diagonal = 10 cm

Example 3: Diagonal of a Rhombus

Problem: Find the other diagonals of a rhombus if it has an area of ​​72cm2 and a diagonal is 9 cm.

Solution:

• Area = 72cm2, a diagonal q=9cm

• Formula: Other diagonal p=2×Areaq

• Substitute: p=2×729=1449=16cm

• Other diagonal = 16 cm

Example 4: Space Diagonal of a Cube

Problem: Find a space diagonal of a cube that has a 5 cm edge.

Solution:

• Edge, a = 5 cm

• Formula for space diagonal: Diagonal=a3

• Substitute: a=5

• d=53

• 3≈1.732→d=5×1.732=8.66cm

• Space diagonal = 8.66 cm

Example 5: Space Diagonal of a Cuboid

Problem: Find the space diagonal of a cuboid whose length is 7 cm, breadth is 4 cm, and height is 3 cm.

Solution:

• Length, l = 7 cm; Breadth, b = 4 cm; Height, h = 3 cm

• Formula: Diagonal = l2+b2+h2

• Substitute: d=72+42+32=49+16+9=74

• d≈8.60cm

• Space diagonal = 8.60 cm

FAQs on Diagonals

1. What shapes have diagonals?

Polygons like squares, rectangles, rhombuses, pentagons, and hexagons, and 3D shapes like cubes and cuboids have diagonals.

2. How many diagonals does a square have?

A square has 2 diagonals that are equal in length.

3. Do all polygons have diagonals?

No, polygons with fewer than 4 sides, like a triangle, do not have diagonals.

4. What is the difference between a face diagonal and a space diagonal?

• Face diagonal: A diagonal on a single face of a 3D shape.

• Space diagonal: A diagonal that passes through the inside of a 3D shape, connecting opposite corners.

5. How do diagonals help in shapes?

Diagonals divide shapes into smaller triangles and help find areas or make 3D shapes stronger.