Diagonal in Geometry

What is Diagonal? 

A diagonal, in geometry, is a line segment joining two vertices of a polygon or polyhedron, if the vertices present are not present on the same edge. In general, any sloping line is called a diagonal.

Here you can read about the basic properties of diagonals, different types of diagonals, and some easy examples. 

Key Definitions to Remember While learning Diagonals

• Polygon: A polygon can be defined as a two-dimensional closed shape with straight sides. It is also defined as a flat or plane. 

There are two types of Polygons are Regular polygon and irregular polygon 

• Vertex: (Plural-vertices) it is described as the corner of any shape.

Number of Diagonals in a Polygon

The number of diagonals in a polygon is based on the number of sides of that particular polygon. We can easily calculate the number of diagonals in a polygon by using a simple formula. 

Number of diagonals = (n(n-3))/2, where n is the number of vertices in the polygon 

For Example- Calculate the number of diagonals in a 12-sided regular polygon. 

According to the formula, number of diagonals in a polygon = (n (n-3))/ 2.

So, the 12-sided polygon will contain 12(12-3)/2 = 54  diagonals.

Diagonal of a Polygon Formula 

If any vertex is connected by sides to two other vertices those connections cannot count as diagonals. That vertex cannot connect to itself either. 

n: the number of sides

sum of diagonal = [(5-3) *5] / 2 = 5

Diagonal of a Square

A square has two diagonals. The line segment joining two corners of a square is known as its diagonal. 

Properties of the Diagonals in a Square

• The diagonals of a square bisect each other. 

• Each diagonal of the square divides the square in such a way that it becomes an isosceles triangle. The isosceles triangles formed are congruent to each other. 

• The diagonals of a square are parallel and perpendicular to each other. 

• In case a circumcircle is drawn, the diameter of the circumcircle is equal to the length of the diagonal. 

• The diagonals of a square are also called congruent diagonals as the length of the diagonals of the square is the same. 

If you have the length of the side of a diagonal then you can easily find out the length of the diagonal of the square. You have to use the given formula to find out the length of the diagonal of a square. 

The length of the diagonal of a square = a/sqrt(2) where a is the length of any one of the sides of the square. 

Diagonal of a Rectangle

A special type of parallelogram whose angles are at ninety degrees is called a rectangle. The diagonals of a rectangle are congruent to each other and they bisect each other. A rectangle is one of the Quadrilaterals with congruent diagonals.  The opposite angles formed at the point where diagonals meet are also congruent.

You can calculate the length of the diagonal of a rectangle easily if you are provided with the dimensions of its sides. The formula given here is the way you can calculate the length of the diagonal in a rectangle. 

The length of the diagonal of a rectangle = sqrt (a² + b²), where ‘a’ is the length of the rectangle and ‘b’ is the breadth of the rectangle.

Rectangle and Formulas 

Area= Length * Width

Perimeter= 2L + 2W

Diagonal = \[\sqrt{L^{2}+W^{2}}\]

Diagonal of a Rhombus

In a rhombus the angles which are opposite to each other are equal. These angles formed are bisected by the diagonals of a rhombus. A rhombus has four sides with two Diagonals. When all the angles of the rhombus turn 90 degrees it becomes a square. 

You can also calculate the length of the diagonal of a rhombus if you are provided with the area. 

A= (PQ)/2, where p and q are the lengths of diagonals of the rhombus. 

The formula for the length of one diagonal of a rhombus is = p = 2a/q

Area of Rhombus 

Area of Rhombus = \[\frac{(d1\ast d2)}{2}\] 

Diagonal d2= \[\sqrt{4a^{2}+d1^{2}}\]

Diagonal d1 = \[\sqrt{4a^{2}+d2^{2}}\]

Diagonal of a Parallelogram

A parallelogram is a type of rectangle whose angle is not equal to 90 degrees. The two diagonals of a parallelogram bisect each other, and the opposite sides and angles of any parallelogram are congruent.

You can also calculate the length of one diagonal of a parallelogram if the dimensions of its sides and the length of the other diagonal is provided to you. 

Length of one diagonal of a Parallelogram = p = sqrt (2a² + 2b² – (q)²) where p is the diagonal length which you don't know, q is the diagonal length provided to you, a

and b is the length and breadth of the parallelogram, respectively.

Diagonal of Parallelogram 

Relation between sides and diagonals of a parallelogram 

Type equation here.

d12 + d22  = 2(a2+b2)