# Diagonal

A diagonal is a line segment that joins one corner to lớn another but is not an edge.So we get a diagonal when we directly join any two corners (vertices) which are not already joined by an edge. In the case of a polygon,it is a straight line connecting the opposite corners of apolygonthrough itsvertices.

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 1 What is a Diagonal? 2 Diagonals of Polygons 3 Number of Diagonals Formula 4 Length of a Diagonal 5 FAQs

## What is a Diagonal?

A line segment joiningone corner khổng lồ another but is not an edge is called a diagonal.So we get a diagonal by directly joining any two corners (vertices) which are not already joined by an edge.

### Shape of a Diagonal

Since a diagonal is a line segment joining non-adjacent vertices or corners, the shape of a diagonal is that of a straight line.

"A diagonal of a polygon is a line segment that is obtained by joining any two non-adjacent vertices." We know that a polygon is a closed shape formed by joining the adjacent vertices. For example, a square has 4 sides, a pentagon has 5sides, và a hexagon has 6sides, và so on. Depending upon the type of polygon basedon the number of edges, the number of diagonals & their properties would vary. Similarly, the properties of diagonals vary according to the type of solid.

Let us now underst& the diagonals for different polygons.

Diagonalof a TriangleDiagonalof a SquareDiagonal of a RectangleDiagonal of a RhombusDiagonal of a PentagonDiagonal of a Hexagon

### Diagonal of a Triangle

A triangle is defined as a closed figure orshapethat has 3 sides, 3angles,& 3 vertices.A triangle is the simplest type of polygon.No vertices in atriangle are non-adjacent. It means that there are no line segments that can khung diagonals.

The number of diagonals of a triangle = 0.

### Diagonal of a Square

A square is defined asa closed two-dimensional figure havingfour sides & four corners. All the sides are parallel lớn each other with equal lengths.The diagonal of a square is a line segment that joins any two of its opposite vertices. In the following square,there are two pairs of non-adjacent vertices. By joining the vertices of each such pair, we get two diagonals, AC and BD of the square. The lengths of the lines AC & BD in the given square are the same. Thediagonal of any square cuts it inlớn two equalright triangles, such that diagonal makeshypotenuseof theright trianglesso formed.The number of diagonals of a square = 2.

### Diagonal of a Rectangle

The diagonal of a rectangle is a line segment that joins any two of its non-adjacent vertices. In the following rectangle, AC và BD are the diagonals. You can see that the lengths of both AC & BD are the same. A diagonal cuts a rectangle into 2 right triangles, in which the sides equal to the sides of the rectangle và with a hypotenuse. That hypotenuse is the diagonal.

### Diagonal of a Rhombus

The diagonals of a rhombus aretheline segments joining the opposite vertices,bisecting each otherat a 90° angle, which means that the two halves of any diagonal will be of the same length.A rhombus can be defined as a diamond-shaped quadrilateral having all four sides equal. The diagonals of a rhombus will have sầu different values unless the rhombus is a square. ### Diagonal of a Pentagon

A pentagon is a closed two-dimensional figure with fivesides & fivecorners. The length of all fivesides is equal in a regular pentagon. A pentagon has five diagonals as shown in the image below: ### Diagonal of a Hexagon

A hexagon is a closed two-dimensional figure with sixsides và sixcorners. The length of all sixsides is equal in a regular hexagon. A hexagon has ninediagonals as shown in the image below: Here, all the 9lines inside the hexagonare the diagonals.

Apart from the polygons, for different solids also, basedon the number of edges, the number of diagonals & their properties would vary.

Diagonal of a CubeDiagonal of a Cuboid

### Diagonal of a Cube

A cube is a three-dimensional solid figure, also known as the square solid that has edges of all the same length. That means that the length, width, and height are equal, & each of its faces is a square. The main diagonal of a cube is the line segment that cuts through its center, joining the opposite vertices.While the diagonal of a face of a cube is the one joining the opposite vertices on every face. Thus, itis not the main diagonal. ### Diagonal of a Cuboid

A cuboid is a three-dimensional analog of a rectangle in two dimensions.The main diagonal of a cuboid is the one that cuts through the center of the cuboid; the diagonal of a face of a cuboidis not the main diagonal. ## Number of Diagonals Formula

The number of diagonals formula can be used khổng lồ calculate the number of diagonals in a polygon. It differs according lớn the type of polygon, based on the number of sides. We can use this formula to find the number of diagonals of any polygon without actually drawing them:

The number of diagonals of a polygon with "n" number of sides =n(n-3)/2

The following table showsthe number of diagonals of some polygons which iscalculated using this formula.

ShapeNumber of sides, nNumber of Diagonals
Triangle3

3(3−3)/2 = 0

4(4−3)/2 = 2

Pentagon5

5(5−3)/2 = 5

Hexagon6

6(6−3)/2 = 9

Heptagon7

7(7−3)/2 = 14

Octagon8

8(8−3)/2 = 20

Nonagon9

9(9−3)/2 = 27

Decagon10

10(10−3)/2 = 35

Hendecagon11

11(11−3)/2 = 44

Dodecagon12

12(12−3)/2 = 54

Example:Find the number of diagonals of a decagon.

Solution:

The number of sides of a decagon is n = 10. The number of diagonals of a decagon is calculated using:

n(n-3)/2= 10(10-3)/2 = 10(7)/2 = 70/2 = 35

The number of diagonals of a decagon = 35

## Length of a Diagonal

The length of a diagonal for any polygondepends upon the type of polygon.There is no general formula lớn calculate the length of a diagonal. Rather, based on the dimensions of the particular polygon, the formula khổng lồ calculate the length of the diagonal can be found. This section will cover the formula lớn calculate the length of diagonal for somepolygons và solids based on their structure and dimensions.

Length of diagonal of a squareLength of diagonal of a rectangleLength of diagonal of a cubeLength of diagonal of a cuboid

### Length of Diagonal of Square

In a square, the lengthof both the diagonals is the same. The length of a diagonal dof a square of side length x units is calculated by the Pythagoras' theorem. Using Pythagoras theorem, d = √(x2+ x2) = √(2x2) =√2x units.

Length of a diagonal of a square =√2x units ### Length of Diagonal of a Rectangle

Similar to a square, the lengthof both the diagonals in a rectangle is the same. The length of a diagonal 'd' of a rectangle whose length is 'l' units andbreadth is 'b' units is calculated by the Pythagoras theorem.

Using Pythagoras theorem, d2 = l2+ b2

Length of a diagonal of a rectangle = √(l2+ b2) units

### Length of a Diagonal of a Cube

Consider a cube of length x units.A cube has6faces. Each face of a cube is a square. Thus each facehas two diagonals. Hence, the length of each such diagonal is the same as the length of adiagonal of a square. Length of each face diagonal of cube =√2x units. Apart from the diagonalson the faces, there are 4other diagonals (main diagonals ortoàn thân diagonals) that pass through the center of the square. The formula for the length of thediagonal of a cube is derived in the same way as we derive sầu the length of the diagonal of a square. Length of bodydiagonal of a cube = √3x units. ### Length of a Diagonal of a Cuboid (Rectangular Prism)

Consider a cuboid of length l, width w,và height h. Let us assume that it'smain diagonal (or body toàn thân diagonal) that passes through the center of the cuboid is d. Length of a diagonal of a cuboid = √(l2+ w2+ h2). Important Notes

The number of diagonals of a polygon with n sides is n(n-3)/2.The length of the diagonal of a square of side xis 2.The length of the diagonal of a rectangle of sides l và b is √(l2+ w2).The length of the diagonal of a cube of side x is√3x.The length of the diagonal of a cuboid of sides l, w, and h is √(l2+ w2+ h2).