Diagonals

A diagonal is an important concept in geometry. A diagonal is a line segment connecting two non-adjacent vertices of a polygon. Diagonals help in dividing polygons into smaller triangles, calculating areas, and understanding the internal structure of shapes. This article will discuss the diagonal, its meaning and more.

Diagonal Definition

A diagonal is a line segment that connects any two non-adjacent vertices of a polygon. It is different from the sides of the polygon because sides always connect adjacent vertices. To draw a diagonal, pick any vertex of the polygon and connect it to another vertex that is not directly adjacent to it. This will form a diagonal.

Here, ABDC is a polygon and BC is a diagonal of the given polygon.

Examples of Diagonals

Below given are a few examples of diagonals.

Diagonals in a Triangle

A triangle has no diagonals because every vertex is connected to the other two vertices by sides.

Diagonals in a Quadrilateral

A quadrilateral, like a square or rectangle, has two diagonals. These diagonals can divide the shape into four right triangles or two congruent triangles.

Applications of Diagonals

Below given are a few applications of diagonals.

• Diagonals are used in dividing polygons into smaller, manageable shapes, making it easier to calculate areas and understand the internal angles.
• Diagonals are also used in architectures. It helps in designing stable structures, such as beams in buildings, where the diagonal supports help distribute weight.
• Diagonals are used in perspective drawing to create depth and dimension.

Diagonals Formulas

To find the number of diagonals in a polygon, we can use the below given formula.

Number of diagonals = n(n-3)/2

where n represents the number of vertices (or sides) of the polygon.

For Example: Calculating Diagonals of a Square

To calculate the number of diagoals of square, which has 4 vertices, we use n = 4 in the above-mentioned formula.

Therefore,

Number of diagonals = 4 (4 – 3)/2

Number of diagonals = 4 (1)/2

Number of diagonals = 2

Therefore, the number of diagonals of the square is equal to 2.

So, the square has 2 diagonals.

General Examples

This formula can be applied to any polygon to find the number of diagonals. For example:

• A triangle has no diagonals.
• A pentagon has 5 diagonals.

Diagonals of Shapes

Diagonals play an important role in the geometry of different shapes. Below discussed are the diagonals of various common shapes.

1. Diagonals of a Triangle

A triangle is a polygon which has three sides and three vertices. A diagonal is defined as a line segment connecting two non-adjacent vertices, and a triangle has no such non-adjacent vertices. Therefore, a triangle does not have any diagonals.

Number of Diagonals in a Triangle: 0

2. Diagonals of a Square

A square has four sides and four vertices. The diagonals of a square connect opposite vertices and have several important properties:

• A square has 2 diagonals.
• The diagonals are congruent. It means that both the diagonals are equal in length.
• The diagonals bisect each other at right angles (90 degrees).
• Each diagonal divides the square into two congruent isosceles and right triangles.

3. Diagonals of a Rectangle

Similar to a square, a rectangle has four sides and four vertices. The diagonals of a rectangle also connect opposite vertices and share similar properties with those of a square:

• A rectangle has 2 diagonals.
• The diagonals are congruent.
• The diagonals of a rectangle bisect each other. They divide the rectangle into two congruent right triangles.

4. Diagonals of a Rhombus

A rhombus is a four-sided polygon (quadrilateral) where all sides have equal length. The diagonals of a rhombus have unique properties:

• A rhombus has 2 diagonals.
• The diagonals bisect each other at right angles.
• The diagonals are not congruent, unlike in a square or rectangle.
• The diagonals divide the rhombus into four right triangles.

5. Diagonals of a Parallelogram

A parallelogram is a polygon with four sides. All the opposite sides of a parallelogram are parallel and equal in length. The diagonals of a parallelogram also have distinct properties:

• A parallelogram has 2 diagonals.
• The diagonals bisect each other but are not necessarily congruent.
• They divide the parallelogram into two congruent triangles.

6. Diagonals of a Pentagon

A pentagon is a polygon with five vertices and five sides. Below given are the properties of the diagonals of a pentagon.

• A pentagon has 5 diagonals.
• These diagonals connect non-adjacent vertices, creating various internal shapes within the pentagon.

7. Diagonals of a Hexagon

A hexagon is a six-sided polygon with six vertices. The number of diagonals in a hexagon increases as the number of sides increases.

• A hexagon has 9 diagonals.
• These diagonals connect non-adjacent vertices, forming various internal triangles and other polygons.

8. Diagonals of a Cube

A cube is a three-dimensional shape with six square faces, eight vertices, and twelve edges. Diagonals in a cube are categorized as face diagonals and space diagonals:

1. Face Diagonals: Each square face of the cube has 2 diagonals.
2. Space Diagonals: A space diagonal passes through the interior of the cube, connecting opposite vertices. A cube has 4 space diagonals.

9. Diagonals of a Cuboid

A cuboid is a three-dimensional shape with rectangular faces. It has face diagonals and space diagonals similar to a cube.

• Face Diagonals: Each rectangular face of the cuboid has 2 diagonals.
• Space Diagonals: There are 4 space diagonals connecting opposite vertices.

Number of Diagonals in Polygons

We know that the formula to calculate the number of diagonals in a polygon is given by:

Number of diagonals = n (n – 3)/2

where n is the number of vertices in the polygon.

Below given table gives us the data for different shapes.

FAQs on Diagonals

What is a diagonal?

A diagonal is a straight line that connects two non-adjacent vertices (corners) of a polygon, cutting across the shape.

How many diagonals does a heptagon have?

A heptagon is a shape which has 7 vertices. It has 14 diagonals.

Does a circle have a diagonal?

No, a circle does not have a diagonal because it does not have vertices or sides.