Diagonal of a Cube Formula: Definition, Formulas, Derivation

Diagonal of a Cube Formula: A cube is a three-dimensional solid object, sometimes known as the square solid, with edges of equal length, breadth, and height, and each face is a square. The diagonal of a cube formula aids in computing the lengths of diagonals of various cubes, such as the face and body diagonals.

In this article, we will understand what is a cube, how a cube’s diagonal is formed, what is a diagonal of a cube, what is face diagonal of a cube formula and more. So, read on!

Cube

A cube is a three-dimensional solid object with six square faces and sides, three of which meet at each vertex. The cube is the only regular hexahedron among the five platonic solids. There are six faces, twelve edges, and eight vertices in it.

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Where, \(L = {\rm{length}},\,B = {\rm{breadth}},\,H = {\rm{height}}\)

Diagonal of a Cube

A cube is a three-dimensional solid shape with all of its edges the same length, also known as the square solid. The object’s length, breadth, and height are the same, and each of its faces is a square. The line segment that goes through the centre of a cube, uniting the opposite vertices, is called the main diagonal. The diagonal of a cube’s face is the line that connects the opposite vertices on each face.

Diagonal of a Cuboid Formula

A cuboid is a three-dimensional equivalent of a two-dimensional rectangle. The diagonal that passes through the centre of a cuboid is the main diagonal; the diagonal of a cuboid’s face is not the main diagonal.

Body diagonal of a cuboid formula \( = \sqrt {{l^2} + {b^2} + {h^2}} \) units

Face diagonal of a cuboid formula \( = \sqrt {{l^2} + {b^2}} \) units, 

where, \(l = {\rm{length}},\,b = {\rm{breadth}},h = {\rm{height}}\)

Diagonal of a Cube Formula

A cube is a \(3D\) shape containing all \(12\) equal edges, it is one of the basic geometric shapes, and we employ cube-shaped objects in real life. Ice cubes, sugar cubes, rubik’s cubes, and playing dice are all examples of cubes in real life. The diagonal of a cube formula aids in determining the dimensions of the face and primary diagonals. The hypotenuse of the right angle is formed by each face diagonal.

Face Diagonal of a Cube Formula

A cube has six square-shaped faces. Each face has two diagonals that connect the non-adjacent vertices. As a result, there are \(12\) face diagonals.

The formula to calculate the length of a face diagonal of a cube is given by,

Length of each face diagonal of cube \( = \sqrt {2a} \) unit, where \(a =\) length of each side of a cube

Body Diagonal of a Cube Formula

The primary diagonals of a cube pass through the cube’s body, connecting the opposite vertices. As a result, there are four diagonals throughout the body.

The formula to calculate the length of a body diagonal of a cube is given by,

Length of each body diagonal of a cube \( = \sqrt 3 a\) unit, where \(a =\) length of each side of a cube

Diagonal of a Cube Formula Derivation

Consider the side of the cube as \(x,\) the face diagonal as \(a\), and the body diagonal as \(d\) to get the diagonal of a cube formula. Right triangles with sides \(x, a,\) and \(d\) are created when the diagonals are joined.

Using the Pythagoras theorem, we have \({x^2} + {a^2} = {d^2} \ldots (i)\)

We know that the face diagonal is \(\sqrt 2 x\) so, \(a = \sqrt 2 x\)

By substituting the value of a in equation (i), we get, 

\({x^2} + {(\sqrt 2 x)^2} = {d^2}\)

\({x^2} + 2{x^2} = {d^2}\)

\(3{x^2} = {d^2}\)

\(d = \sqrt {3{x^2}} \)

\(d = \sqrt 3 x\)

Therefore, the diagonal of a cube formula \( = \sqrt 3 x\) units

Number of Diagonals in a Cube

A cube has six faces, with two diagonals connecting non-adjacent vertices on each face and four diagonals running through the cube’s body.

 In a cube, there are a total of \(12+4=16\) diagonals.

Use of Diagonal of a Cube

When measuring the diagonal of the cube’s square-shaped faces, the diagonal of a cube formula comes in handy. The face diagonal and the cube’s main diagonal are calculated using the cube’s diagonal formula. The face diagonal is the cube’s side diagonal. Therefore \(6\) faces equal \(12\) diagonals.

Solved Examples Using Diagonal of a Cube Formula

Q.1.  Calculate the length of the diagonal of a cube whose each side measures 6 units.
Ans:  From the given, \(a=6\) units
The length of body diagonal of a cube formula \( = \sqrt 3 a\) unit, where \(a =\) length of each side of a cube
\( = \sqrt 3 \times 6\)
\( = 1.732 \times 6\)
\( = 10.392\) units
Hence, the obtained length of the diagonal of a cube is \(10.392\) units

Q.2. The length of the diagonal of a cube is \(9 \sqrt{3}\) Units. Find the length of each side.
Ans: From the given information, we can say that the length of each body diagonal of a cube \( = \sqrt 3 a\) unit, where \(a =\) length of each side of a cube
\( \Rightarrow 9\sqrt 3 = \sqrt 3 \times a\)
\( \Rightarrow a = \frac{{9\sqrt 3 }}{{\sqrt 3 }}\)
\( \Rightarrow a = 9\) units
Hence, the obtained length of each side of a cube is \(9\) units.

Q.3.  Find the length of the face diagonal of a cube when the side measures 9 units. Use the diagonal face formula of a cube. 
Ans: From the given information, we get, \(a=9\) units
Now, the length of each face diagonal of a cube \( = \sqrt 2 a\) unit, where \(a =\) length of each side of a cube
\( = \sqrt 2 \times 9\)
\( = 1.414 \times 9\)
\( = 12.726\) units
Hence, the obtained length of the face diagonal of a cube is \(12.726\) units.

Q.4. The side length of a cube is 5 meters. What is the length of its diagonal across one of the faces?
Ans: We know all sides of a cube are equal, and all the sides form right angles. By using the Pythagoras theorem, we can get the length of the diagonal.
\({x^2} + {a^2} = {d^2}\)
\( \Rightarrow {a^2} + {a^2} = {d^2}\)
\( \Rightarrow {5^2} + {5^2} = {d^2}\)
\(\Rightarrow 25 + 25 = {d^2}\)
\( \Rightarrow {d^2} = 50\)
\( \Rightarrow d = \sqrt {50} \)
\(\Rightarrow d=\sqrt{25 \times 2}\)
\(d=5 \sqrt{2}\)
Hence, \(5 \sqrt{2} \) is the length of its diagonal across one of the faces.

Q.5. The side length of a cube is 5 cm. Find the diagonal of a cube with this length of the side.
Ans: From the given, \(a = 5\;{\rm{cm}}\)
So, we have the length of each body diagonal of a cube \( = \sqrt 3 a\) unit, where \(a =\) length of each side of a cube
\( = \sqrt 3 \times 5\)
\( = 1.732 \times 5\)
\( = 8.66\;{\rm{cm}}\)
Hence, the obtained length of the diagonal of a cube is \(8.66\;{\rm{cm}}\)

Summary

A cube, commonly known as the square solid, is a three-dimensional solid object with all of its edges the same length. The primary diagonal is a line segment that runs through the centre of a cube, connecting the opposite vertices. The line connecting the opposite vertices of a cube’s face is called the diagonal.  This article includes the definition of the diagonals of a cube, its formulas, derivation and uses. 

This article helps in better understanding of the topic diagonals of a cube. The outcome of this article helps in applying the suitable formulas while solving the various problems based on them.

FAQs on Diagonal of a Cube Formula

Frequently asked questions related to diagonals of a cube formula are listed as follows:

Q.1. What is a diagonal of a cube?
Ans: A cube is a three-dimensional solid shape with its edges the same length, also known as the square solid. The object’s length, breadth, and height are the same, and each of its faces is a square. The line segment that goes through the centre of a cube, uniting the opposite vertices, is called the main diagonal. The diagonal of a cube’s face is the line that connects the opposite vertices on each face.

Q.2. What is the formula of the diagonal of a cube?
Ans: The formula to calculate the length of a face diagonal of a cube is given by:
The length of each face diagonal of cube \( = \sqrt 2 a\) unit, where \(a =\) length of each side of a cube.
The formula to calculate the length of a body diagonal of a cube is given by:
Length of each body diagonal of a cube \( = \sqrt 3 a\) unit, where \(a =\) length of each side of a cube.

Q.3. Give an example of the diagonal of a cube.
Ans: Ice cubes, sugar cubes, Rubik’s cubes, and playing dice are all examples of cubes in real life. The diagonal of a cube formula can calculate the dimensions of the face and primary diagonals. The hypotenuse of the right-angled triangle created is formed by each face diagonal.

Q.4. Are the diagonals of a cube equal?
Ans: A cube has two diagonals, one is the face diagonal, and the other is the main diagonal. The lengths of these two diagonals are not equal. The length of the main diagonal is always greater than the face diagonal.

Q.5. What is the main diagonal of a cube?
Ans: The diagonal that passes through the centre of the cube is the main diagonal; the diagonal on the cube’s face is not the main diagonal. Every cube major diagonal can be calculated by multiplying one side’s length by the square root of three.

Q.6. What is the diagonal of a square?
Ans: A square’s diagonal is a line segment that connects two non-adjacent vertices. A square has two equal-length diagonals that bisect each other at right angles.

Q.7. What is the formula of the diagonal of a cuboid?
Ans: Body diagonal of a cuboid formula \(=\sqrt{l^{2}+b^{2}+h^{2}}\) units
Face diagonal of a cuboid formula \(=\sqrt{l^{2}+b^{2}}\) units,
where, \(l = {\rm{length}},\,b = {\rm{breadth}},h = {\rm{height}}\)

Study Volume of Cube Formula

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