Diagonal Formula

Diagonal formula determines the number and length of diagonals present in a polygon. A diagonal is a slanted line segment that joins any two vertices of a given polygon. These are straight lines that connect two corners of a square, rectangle, rhombus, or another polygon with more than three sides.

Students can learn diagonal formula and the examples based on them in detail by referring to the Extramarks website or mobile application.

What is a Diagonal?

Diagoanl is any line segment that joins two vertices of a polygon that are not previously connected by an edge.

• Diagonals are different from the edges as they do not determine the boundary of a polygon.
• The diagonals are always in the form of a straight line.
• Simply we can say that, a diagonal is a line that, at its vertex, joins the opposite corners of a polygon or a polyhedron.
• Diagonal is always located inside the interior of convex polygons.
• Concave polygons may have diagonals that cross sides and partially lie outside

NOTE: A triangle does not have any diagonal. Diagonal is always present in a polygon with atleast four sides.

What is Diagonal Formula?

Diagonal Formula is a mathematical formula used to calculate the number and of diagonals present in a polygon. We know that when we join two opposite vertices of a polygon we get a diagonal. However, the number of diagonals and its length depends upon the type of polygons. We can also calculate diagonal lenth using distance formula which we study in higher class, however there are some direct diagonal formulas that are used to calculate diagonal length. Let’s learn them below in detail.

Formula for Diagonals

In order to determine how many and how long diagonals are present in a polygon, one can use the Diagonal Formula. The formula for determining how many diagonal lines there are in an n-sided polygon is as follows:

Number of Diagonals Present in a Polygon = n(n – 3)/2

here,

n is the number of sides of the polygon

There are various Diagonal Formula in geometry for the calculation of length of diagonals in various types of polygons. These calculations are utilised only for that specific polygon.

Diagonal of a Square

In order to determine the length of a diagonal, let’s now examine a few different Diagonal Formula.

Diagonal of square = a√2

here

a is the length of the side of the square.

Diagonal of a Rectangle

For rectangles, the diagonal formula is given as

Diagonal of a Rectangle = √(l² + b²)

here,

• l is length of rectangle
• b is breadth or width of rectangle

Diagonal of a Parallelogram

In a parallelogram, the diagonals are of unequal length. If p and q are length of diagonals for a Parallelogram ABCD, then its diagonal formula is given as

• p = √(a² + b² – 2ab.cos A)
• q = √(a² + b² + 2ab.cos A)

here,

• p and q are diagonals
• a and b are adjacent sides
• A is angle of Parallelogram

NOTE: If angle A is given for parallelogram ABCD and we need to use angle B in above formula we can find by the formula (angle B = 180 – angle A)

If one diagonal’s length is known, the formula for the other diagonal can be calculated from the below formula

p² + q² = 2(a² + b²)

Where

• a and b are the sides of the parallelogram
• p and q are the two diagonals of the parallelogram

Diagonal Formula of Rhombus

The diagonal formula for Rhombus is given as

• p = 2A/q
• q = 2A/p

here,

• p and q are length of diagonals
• A is area of rhombus

Examples Using Diagonals Formula

Example 1: Sam is walking around a rectangular park whose length of 10m and breadth is 8m. Find out the diagonal of a rectangular park where Sam is walking.

Solution:   

To find The diagonal of a rectangular park.                   

Given parameters are,

Length = 10m

Breadth = 8m

Using the Formula for Diagonals,

Rectangle Diagonal = √[l² + b²]

= √[10² + 8² ]

= √[164]

= 12.80 m

Answer:  The diagonal of a rectangular park where Sam is walking is 12.80 m.

Example 2: The area of the rhombus is 100 square inches. Determine the second diagonal of a rhombus whose one of its diagonal one measures 10 inches.

Solution:  

To find: The second diagonal of a rhombus

Given parameters are,

The area of the rhombus = 100 inch²

Using the Diagonal Formula,

Diagonal of a Rhombus, p = 2(A)/q and q = 2(A)/p         

p = 2(100)/10

p = 20 inches.

Example 3: Determine the length of the diagonal of a square whose side measure is 5 units

Solution:

To find: The diagonal of a square                   

Given parameters are,

Side of square = 5 units

Using the Formula for Diagonals,

Square Diagonal = a√2

= 5√2

= 7.07 units

Answer:  The diagonal of a square is 7.07 units.