✅ Diagonal Of A Square Formula ⭐️⭐️⭐️⭐️⭐

Diagonal of Square

The diagonal of a square is a line segment that joins any two non-adjacent vertices. A square has two diagonals that are equal in length and bisect each other at right angles. The diagonal of square formula is used to calculate the length of the diagonal of a square when its side length is known.

What is the Diagonal of a Square?

A square has two diagonals and each diagonal is formed by joining the opposite vertices of the square. Observe the following square to relate to the properties of the diagonals given below.

• The diagonals of a square are equal in length.
• They are perpendicular bisectors of each other.
• They divide the square into two congruent isosceles right-angled triangles.

What is the Diagonal of Square Formula?

The diagonal of a square formula, is d = a√2; where ‘d’ is the diagonal and ‘a’ is the side of the square. The formula for the diagonal of a square is derived using the Pythagoras theorem. A diagonal divides a square into two isosceles right-angled triangles. Both the diagonals are congruent and they bisect each other at right angles. Let us understand how to derive the formula to find the diagonal of a square.

Derivation of Diagonal of Square Formula

In a square, the length of both the diagonals is the same. The length of a diagonal, ‘d’ of a square with side length ‘a’ is calculated using the Pythagoras theorem. Observe the following square to see that the length of the diagonal is denoted by the letter ‘d’ and the side length is denoted by ‘a’.

Diagonal of a Square Formula

Let us consider the triangle ADC in the square. We know that all the angles in a square are 90°, therefore, using the Pythagoras theorem, we can find the hypotenuse, which is ‘d’ in this case.

d² = a² + a²

d = √(a² + a²)

d = √(2a²)

d = √2 × √a²

= √2a

Therefore, the diagonal of a square formula is: d = a√2

What is the Formula for the Diagonal of a Square?

Diagonal Of A Square Formula

In geometry, a square is a regular quadrilateral that has four equal sides and four angles which are 90∘90∘ in measurement. Thus the sum of all the internal angles of a square is equal to 360∘360∘.

Diagonal of a square-

A square can have two diagonals. Each of the diagonal can be formed by joining the diagonally opposite vertices of a square. The properties of diagonals are as follows-

• Both the diagonals are congruent (same length).
• Both the diagonals bisect each other, i.e. the point of joining of the two diagonals is the midpoint of both the diagonals.
• A diagonal divides a square into two isosceles right-angled triangles.

Length of the Diagonal-

Consider a square ABCD having sides equal to ‘a’ cm.

Let AD and BC be the two diagonals of a square.

The Properties of a Square

Diagonal of a Square Formula: Derivation, Area, Half Diagonal

Diagonal of a Square Formula: The line that extends from one corner of the square or rectangle to the opposite corner through the centre of the figure is known as the diagonal. Any square that has two diagonals are equal in length to each other. The diagonal formula is used to calculate the polygon diagonals. Diagonals are a line joining two non-adjacent vertices of a polygon, i.e., a diagonal joins two vertices of a polygon excluding the edges of the figure.

The diagonal of a square is a line segment that connects any two non-adjacent vertices. A square has two diagonals that are equal in length and bisect each other at right angles. The diagonal of a square formula is used to calculate the length of the diagonal of a square when its side length is known.

Diagonal of Square Formula Derivation

In a square, the length of both the diagonals is the same. The length of a diagonal dd of a square of side length x is calculated using the Pythagoras theorem. Observe the following square to see that the diagonal length is denoted by the letter dd and the side length is denoted by x. Let us understand how to derive the formula to find the diagonal of a square.

Diagonal of a Square Formula Area

The magnitude of the measurement of the region enclosed by a closed plane figure is called its area.

A square has two diagonals, and each diagonal is formed by joining the opposite vertices of the square. Observe the following square to relate to the properties of the diagonals given below.
1. The length of the diagonals of a square is equal in measure.
2. The diagonal of a square are perpendicular bisectors of each other.
3. The diagonal of a square split the square into two congruent isosceles right-angled triangles.

Diagonal of a Square Formula Area Derivation

The area of a square using diagonal length is as follows:

Half Diagonal of Square Formula

Diagonal of a Rectangle Formula

Solved Examples – Diagonal of a Square Formula

Frequently Asked Questions – Diagonal of a Square Formula

Q.1. What is the diagonal formula of the square?
Ans: The diagonal of a square is a line segment that joins any two non-adjacent vertices. A square contains two diagonals that are equal in length and bisect with one another at right angles. The diagonal of the square formula is used to calculate the length of the diagonal of a square when its side length is known.

Q.6. What do you understand by the diagonal of a square in Mathematics?
Ans: A square has two diagonals, and each diagonal is formed by joining the opposite vertices of the square. Observe the following square to relate to the properties of the diagonals given below.
1. The length of the diagonals of a square is equal in measure.
2. The diagonal of a square are perpendicular bisectors of each other.
3. The diagonal of a square split the square into two congruent isosceles right-angled triangles.

What is Diagonal?

We have studied what is a polygon. To recall, a two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.

The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal.

A diagonal of a polygon is a line from a vertex to a non-adjacent vertex. So a triangle has no diagonals. You cannot draw a line from one interior angle to any other interior angle that is not also a side of the triangle. A quadrilateral, the next-simplest, has two diagonals. A pentagon has five diagonals. The shapes with a diagonal drawn on them are given below: 

We have a number of diagonals formula of a polygon which help us to know the number of diagonals in a polygon that is 

Where the number of sides of a polygon is n

From this formula, we can easily calculate the number of diagonals in a polygon.

The Given Table Shows the Number of Diagonals in Different Polygons

Let us study diagonal formula for different shapes

Diagonal of Square Formula

Where s is the side of a cube.

Diagonal of Cuboid Formula

FAQs (Frequently Asked Questions)

1. State Diagonals Examples in Real Life.

Answer: Some of the real-life examples of diagonals are:

• Diagonals in squares and rectangles add strength to construction, whether for a house wall, bridge, or tall building. 
• You may see diagonal wires used to keep bridges steady. 
• When houses are being built, look for diagonal braces that hold the walls straight and true.
• Bookshelves and scaffolding are braced with diagonals.
• For a catcher in softball or baseball to throw out a runner at second base, the catcher throws along a diagonal from home plate to second.
• The phone or computer screen is measured along its diagonal.

2. What is the Formula to Find the Number of Diagonals in any Polygon?

Answer: Formula for the number of diagonals

As we know that, the number of diagonals from a single vertex is three less than the number of vertices or sides, or (n-3).

Consider there are n number of sides and n vertices, which gives us n(n-3) diagonals. Each diagonal has two ends, so this would count each diagonal twice. So we divide by 2, for the final formula:

Number of diagonals = n(n−3)/ 2

Where n is the number of sides.

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