What is a Diagonal - Meaning, Examples

What is a Diagonal?

A diagonal is a straight line that connects two corners of a shape, but only if those corners are not next to each other.

For example, a square has four corners. If you draw a line from the top-left corner to the bottom-right corner, you are drawing across the shape. That line is called a diagonal. It is not one of the sides, but it still connects two corners.

What Does a Diagonal Shape Look Like?

A diagonal shape is not a special shape on its own. It means any regular shape, like a square or rectangle, that has one or more diagonal lines drawn inside it.

If you take a square and draw a line from one corner to the opposite corner, that line is a diagonal. It cuts across the shape in a slanted way.

The shape still stays the same, but now it has a diagonal inside it. These diagonals help you divide the shape into smaller parts or understand more about how the corners are connected.

Diagonals of Polygons

Polygons are flat shapes with straight sides. Some common polygons are triangles, squares, rectangles, pentagons, and hexagons. The number of diagonals you can draw depends on how many corners, or vertices, the polygon has.

Here's a formula to calculate diagonals of polygon:

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

(where n is the number of sides)

So, if a polygon has 5 sides (like a pentagon), you can find the number of diagonals by plugging in the number:

5(5 - 3)/2 = 10/2 = 5 diagonals

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Diagonals of a Triangle

A triangle  has three sides and three corners. 

Can we draw any diagonals in a triangle?

No, we cannot. That is because each corner is already connected to the other two. There are no extra corners to connect that are not already joined by a side.

So a triangle has zero diagonals. Even if it seems like you could draw one, all the lines between corners are already sides of the triangle. 

Diagonals of a Square

A square is a shape with four equal sides and four corners. In a square, we can draw diagonals.

There are two diagonals in a square. Each one goes from one corner to the opposite corner. When you draw both diagonals, they cross in the center of the square.

These two diagonals are the same length. They divide the square into four equal right-angled triangles.

Each diagonal also cuts the square into two equal parts. This helps us understand more about the shape and how it can be split or measured.

Diagonals of a Rhombus

A rhombus is a shape that looks like a diamond. It has four equal sides like a square, but the corners are not always right angles.

A rhombus also has two diagonals. Just like in a square, they go from one corner to the opposite corner.

The special thing about the diagonals of a rhombus is that they cross each other in the middle and form right angles. That means they meet at 90 degrees. They also cut each other exactly in half.

Even if one diagonal is longer than the other, they still divide the rhombus into four right-angled triangles. This makes a plus sign in the center of the shape.

Read More: Area of a Rhombus 

Diagonals of a Hexagon

A hexagon is a shape with six sides and six corners. In a regular hexagon, all sides and all angles are equal. Because there are more corners, there are more ways to connect non-adjacent corners.

A hexagon has nine diagonals. These diagonals go from one corner to other corners that are not directly beside it. If you draw all the diagonals inside a hexagon, you will see that it gets divided into many smaller triangles and shapes.

Diagonals of 3d Shapes

Three-dimensional shapes are solid shapes that have length, width, and height. Just like flat shapes have diagonals, 3d shapes alaso have them. In 3d shapes, there are two kinds of diagonals. One type is called a face diagonal and the other is called a body diagonal.

A face diagonal is a line that goes across one flat face of the shape. It connects two opposite corners on the same face. A body diagonal is a longer line that goes through the middle of the solid. It connects one corner of the shape to the farthest opposite corner.

The number and length of diagonals depend on the shape. Let us look at two common 3d shapes that have diagonals. These are the cube and the cuboid.

Diagonals of a Cube

A cube has six square faces, twelve edges, and eight corners. In a cube, there are two types of diagonals. One type is called a face diagonal. The other type is called a body diagonal.

Each face of a cube is a square. On each square face, you can draw two diagonals, just like in a flat square. Since a cube has six square faces and each has two diagonals, there are twelve face diagonals in total.

A body diagonal goes from one corner of the cube to the opposite corner that is farthest away. These diagonals pass through the middle of the cube. There are four body diagonals in a cube.

So, a cube has twelve face diagonals and four body diagonals.

Read More: Cube root

Diagonals of a Cuboid

A cuboid is like a stretched cube. It has six faces, twelve edges, and eight corners. The difference between a cuboid and a cube is that the sides of a cuboid are rectangles, not squares.

Just like a cube, a cuboid has two types of diagonals. Each rectangular face has two face diagonals. Since there are six faces, that gives us twelve face diagonals. A cuboid also has four body diagonals. These go from one corner to the opposite corner through the center of the shape.

So, a cuboid has twelve face diagonals and four body diagonals.

More About Diagonals of Polygons

Let’s explore a few more polygon types to see how diagonals work:

Pentagon

A pentagon has five sides. If you try drawing lines between corners that are not next to each other, you will find that there are five diagonals in total.

Hexagon

A hexagon has six sides. It has nine diagonals. That’s a lot of lines!

Octagon

An octagon has eight sides, like a stop sign. It has twenty diagonals!

All these shapes help us understand how diagonals of polygons increase as the number of sides increases.

Diagonal Examples in Real Life

Now that you know the basic idea, let’s look at some diagonal examples from real life. Diagonals are everywhere!

1. A kite: The wooden sticks that cross in the middle of a kite are diagonals.

2. Window panes: Many windows have metal strips forming a cross across them — those are diagonals.

3. Floor tiles: If you’ve seen square tiles placed in a slanted way, the lines across them are diagonals.

4. Letter X: The two lines in the letter X act like diagonals.

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How to Calculate the Length of a Diagonal

We can even calculate the length of a diagonal in some shapes using simple math.

Diagonal of a Square

If each side of the square is “a”, the diagonal (d) is:

d = a√2

So, if the side of the square is 4 units, the diagonal will be:

d = 4√2 ≈ 5.66 units

This formula is helpful in geometry and construction work.

Diagonal of a Rectangle

If the length is “l” and width is “w”:

d = √(l² + w²)

This uses the Pythagoras theorem, which you will learn in more detail in later grades.

1: A regular pentagon has 5 sides. How many diagonals can be drawn inside it?

Solution
We use the formula:

Number of diagonals = n × (n − 3) ÷ 2

Here, n = 5

5 × (5 − 3) ÷ 2

(5 × 2) ÷ 2

10 ÷ 2 = 5

So, a regular pentagon has 5 diagonals.

2. How many diagonals does a polygon with 10 sides have?

We are given that the number of sides is 10.

We will use the same formula:

Number of diagonals = n × (n − 3) ÷ 2

10 × (10 − 3) ÷ 2

10 × (7) ÷ 2

70 ÷ 2 = 35

A polygon with 10 sides has 35 diagonals.

3. A polygon has 20 diagonals. How many sides does it have?

We will try the formula with different values of n.

Let us try n = 8

Number of diagonals = n × (n − 3) ÷ 2

8 × (8 − 3) ÷ 2

8 × (5) ÷ 2

40 ÷ 2 = 20

This matches the number of diagonals given in the question. The polygon has 8 sides. It is called an octagon.

Also Read: Area of Shapes

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