Diagonal of a Cube Formula

Introduction

A Cube is one of the solid three-dimensional figures. It has 6 square faces, 12 edges, and eight vertices, in geometry. To put it plainly, the shape of this figure is that of a box which we all encounter every now and then every single day. 


You can check In the image below that B stands for width, L stands for length, and H stands for height. There are a number of applications of figures like cubes in our day-to-day life. Not only that. Its horizons extend toward mathematical problems as well, which makes it one of the most significant geometric shapes. 


Properties of Cube:

  • A Cube has 6 faces which are all in a square shape.

  • When it comes to the dimensions of all the faces or sides of a Cube, they are all equal.

  • All the angles of the planes on the Cube are at the right angle.

  • Each of the faces of a Cube meets 4 others at each of the edges. 

  • Every single vertex of a Cube has three edges and three faces.

  • Opposite edges on a Cube are parallel to each other.


More About the Topic

Cube is a solid three-dimensional figure with 6 square faces, eight vertices, and 12 edges, in geometry. It is also said to be a regular hexahedron. In simple words, it is a solid box-shaped object with six identical square faces. In the image below, L stands for length, B stands for width, and H stands for height. There are many applications of cubes in day-to-day life as well as in mathematical problems, thereby making it one of the most important geometric shapes.

 

 

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Here are a few examples of cubes in daily life:

  • Dice

  • Rubik’s Cube

  • Ice and sugar cube

  • Building blocks

  • Boxes

 

Note: Since the cube is a solid figure, it has three dimensions, while the plane figures have only two dimensions.

 

Face Diagonal of a Cube Formula:

A cube has six faces. So, it’ll have twelve face diagonals (two diagonals on each face). The diagonals form the hypotenuse of a right-angled triangle in the Cube, making each leg be formed by one edge. Hence, the length will be (a\[\sqrt{2}\]).

 

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Diagonal of a Cube Formula:

Now, let's discuss the length of the diagonal of the cube formula. The main diagonal of the Cube is the one that cuts through the center of the Cube; the diagonal of the Cube's face is not the main diagonal. The main diagonal of every Cube can be determined by multiplying one side's length with the square root of 3 (this is also known as the body diagonal of a cube formula).

 

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As discussed earlier, in a cube, all sides have equal length.

So, let's say that each side has a length x.

Then, formula to find diagonal of cube = \[\sqrt{3}x\].

 

Diagonal of a Cube Formula Derivation:

Let’s consider cube with face diagonal ‘a’ & body diagonal ‘d’.

 

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In the above diagram, we have a right triangle with sides a, c and d.

By Pythagoras Theorem, we have

a2 + c2 = d2 - (Equation 1)

We already discussed the formula of face diagonal i.e., c = a√2

By substituting the value of c in Equation 1, we get,

\[\Rightarrow a^{2} + (a \sqrt{2})^{2} = d^{2}\]

\[\Rightarrow a^{2} + (2a^{2}) = d^{2}\]

\[\Rightarrow 3a^{2} = d^{2}\]

\[\Rightarrow d = \sqrt{3a^{2}} \]

\[\Rightarrow d = a \sqrt{3} \]

Therefore, Diagonal of cube = \[a\sqrt{3}\].

 

Examples based on Diagonal of a Cube Formula:

Example 1: Find the diagonal of the cube with the given side 10 cm?

Solution: Side is given as, x = 10 cm

We know that,

The diagonal of the cube is \[\sqrt{3}x\].

So, Diagonal of cube = \[\sqrt{3}\] (10)

We know that the value of \[\sqrt{3}\] is 1.734

Therefore, the diagonal of a cube = 1.734 x (10)

So, Diagonal of cube = 17.34 cm

 

Example 2: If the edge of a cube measures 5 cm, find the length of a diagonal.

Solution: Side is given as, x = 5 cm

We know that, 

Diagonal of cube = \[\sqrt{3}x\].

Diagonal of cube = \[\sqrt{3}\] (5)

We know that the value of \[\sqrt{3}\] is 1.734

Therefore, the diagonal of a cube = 1.734 x (5)

So, Diagonal of cube = 8.67 cm.

 

Other Formulas of Cube are as Follows:

The other formulas mentioned for cube here are defined based on their Surface areas (TSA), (LSA) Lateral surface areas, and Volume(V):

Total Surface Area = \[6 \times Side^{2}\]

Lateral Surface Area =  \[4 \times Side^{2}\]

The volume of cube = \[ Side^{23}\]

Perimeter of cube = 12 x Side

 

The Difference Between Cube and Cuboid:

A cube is a three-dimensional shape recognized as an XYZ plane. There are six faces, eight vertices, and twelve edges in it. All the cube faces have equal dimensions and are square in shape.

 

A polyhedron with six faces, eight vertices, and twelve edges is a cuboid. The cuboid faces are parallel to each other. But not all the cuboid faces are dimensionally equivalent.

 

Major differences between both are as follows:

  • The cube sides are the same, but the cuboids are different.

  • In shape, the cube's sides are square, but the cuboid is rectangular.

  • All the cube diagonals are the same, but the cuboid has the same diagonals for parallel sides only.

  • These are the key points that are the reasons why there is a difference between the formula of diagonal of cube and cuboid.

 

Why is a Cube a Special Kind of Cuboid?

A cube can be considered as a special kind of cuboid where all the cuboid faces are of equal length. There are six faces in the cuboid, which are rectangles. If the rectangles have the same sides, they become squares, and eventually, the cuboid becomes a cube.

 

Conclusion

These notes are prepared in such a way that it's easy to understand the concepts of cubes and cuboids. We observed that a 3D solid cube has six sides. There are 8 vertices and 12 edges so that at one vertex, three edges meet. Here, we also derived the formula for the diagonal of a cube and compared the features of cubes and cuboids. So, you must go through the solved examples given here to understand and learn the concepts and their applications well.

FAQs on Diagonal of a Cube Formula

1. How Will You Find the Diagonal of a Cube?

The diagonal of a cube is the line that goes through the centre of it. One can easily find the diagonal of the face of the cube, but that is not the main diagonal of a cube. In order to find the diagonal of a cube, one can determine it by multiplying the length of a side of the cube, which are all the same with the square root of 3. This easy calculation can help you take out a cube’s diagonal very easily. 

2. What is the Diagonal of a Cube?

As we talk about the diagonal of a cube, it must be known that it is a section of the polygon that joins two points within the cube. These points are not basically the endpoints of one edge. In fact, you must know that there are 2 diagonals of a cube. One is what passes through the square and the other is what passes through the cube. Whereas the two diagonals of the square are of the same length, the cube has diagonals of two different lengths, the shorter ones on the square and the longer ones passing through the centre.

3. What is the Face Diagonal of a Cube?

A cube, as we all know, has 6 faces. Out of these 6 faces, it consists of 2 diagonals on each face. So, it can be determined by multiplying 6, for faces, by 2, for diagonals on each face, that there are a total of 12 face diagonals of a cube. Each of the diagonals that are formed is hypotenuses, meaning that they make for a right-angled triangle. It is also easy to find their length using the Pythagorean theorem. 

4. What is the angle of the diagonal of a cube?

A cube has sides that are all the same in length. When it comes to the diagonals too, it can be determined that their angle is 90 degrees. In order to prove this, you can ensure to have the lengths of the sides with you. When you do, there is a chance to figure out what the length of the diagonal is as well. You can, very easily, use the Pythagoras Theorem and find the actual lengths of the diagonals of a cube. 

5. How can Pythagoras Theorem be used as part of taking out the length of a diagonal of a cube?

A cube,  as we all know, has the length of all its edges the same. When one needs to take out the length of a diagonal, it is very easy. All you have to do is ensure to know the correct length of the sides. Now, since the sides are all the same in length, you can use the Pythagoras theorem, because the 2 edges of the cube and a diagonal on the same face make for a right-angled triangle. This can be a very easy process with the right use of the Pythagoras theorem as well as the knowledge of the lengths of the edges.  

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