Introduction

The diagonal of a cube is nothing but a line segment that connects any two distant apexes of a cube. Moreover, a cube is basically a 3D object that is restricted by six square faces, sides with a number of three meeting at each and every pinnacle. A cube is found to hold two different types of diagonals, one is residing on square faces as well as another is going through the center of the cube.   

Discussion

What is Cube-Diagonal?

A cube is one the most usually ocular 3D geometric shapes around us with a number of 12 equal edges. Moreover, there are six square faces by which a cube is restricted. It can be said that the diagonal of a cube is basically a line segment that is liable to adjoin the two different points that must not be the finish points of an edge among the 12 equal edges of a cube. Moreover, in this regard, it can be stated that there are two different diagonals with different lengths present in a cube. One diagonal which is the shorter one is found to reside on square faces as well as the longer diagonal is found to go through the center point of a cube. In this regard, it can be said that the diagonal which goes through the center of a cube as well as connecting the opposite vertices is called the main diagonal.

Types of Cube-Diagonal

Due to the three-dimensional shape, there are two kinds of diagonals present in a cube. The types are listed below.

Face Diagonal of a cube 

A cube is basically a 3D geometric shape that has a number of six square-shaped faces. Moreover, each and every square-shaped face is liable to have two diagonals that are basically joining the two distant vertices. These are called face cube-diagonals as well as there are 12 face cube-diagonals present in a cube. Moreover, the face diagonals of a cube are the shorter-length diagonals.  

Body diagonal of a cube 

On the other hand, there are also some diagonals present in a cube that are liable to append the converse vertices of the cube going through the center of the cube. These are basically called the body diagonals or can say space diagonals of a cube. There are a number of four body diagonals present in a cube that are liable to connect the converse corners of a cube via going through the center. Moreover, the body diagonals of a cube are the longer-length diagonals.

Formula of Cube-Diagonal

The formula by which the length of a face diagonal of a cube can be calculated is √2a, where a is the length of each and every side of a cube.

The formula by which the length of a body diagonal or can say a space diagonal of a cube can be calculated is √3a, where the a is the length of each and every side of a cube.  

Derivation of Cube-Diagonal formula

With the help of the Pythagoras theorem, the cube-diagonal formula that is usually the body diagonal formula of a cube can be derived.

Let’s consider, a= side of a cube, r= a face cube-diagonal as well as d= a body cube-diagonal. After connecting the face cube-diagonal and the body cube-diagonal, a right-angled triangle is observed which has sides a, r as well as d as the hypotenuse. Therefore, by applying the theorem of Pythagoras, an equation is formed. That is a2 + r2 = d2 . As the face cube-diagonal formula is √2a, thus, in place of r, one can write √2a. 

a2 + (√2a)2 = d2

a2 + 2a2 = d2

3a2 = d2

d = √(3a2)

d = √3a

In this way, the body diagonal of a cube that is √3a can be obtained.

Uses of Cube-Diagonal

The formula of the diagonal of a cube can be useful while calculating the diagonal of the square-shaped faces of a cube. The face diagonal as well as the main diagonal that is the body diagonal of the cube can be calculated by using the cube-diagonal formula.

Conclusion 

In order to conclude the whole discussion, the cube-diagonal formula is used to calculate the different kinds of diagonals in a cube. Moreover, by using the theorem of Pythagoras, the formula of the main diagonal, or can say body diagonal of a cube can be derived. Thus, a clear understanding of cube-diagonal has been gained in this discussion in terms of types and uses.