Diagonal of a Cube Formula
Last Updated :
08 Aug, 2024
Diagonal of a cube is the line segment joining the two non-adjacent vertices of a Cube. The diagonal of a cube formula helps us to calculate the length of diagonals in a cube. There are primarily two diagonals in a cube, namely face diagonals and body diagonals. In this article, we will learn the types of diagonals in a cube, the diagonal of a cube formula, the derivation of the diagonal of a cube formula, and problems based on it.
What is a Diagonal of a Cube?
A diagonal of a cube is a line segment that joins the two opposite vertices of a cube. In general, a diagonal is a line segment that joins two opposite vertices in a polygon having more than three sides. Diagonals are not present in a triangle. A Cube is a three-dimensional solid figure whose all three dimensions i.e. length, breadth, and height are equal in measurement. Some of the examples of cubes that we can see in our daily life are ice cubes, Rubik's cube, and the die of Ludo, etc. A Cube has 6 faces, 12 edges, and 8 vertices. Apart from faces, edges, and vertices, there are diagonals inside a cube.Â
There are two types of diagonals in a cube:
- Face Diagonal of a Cube
- Body Diagonal of a Cube
Let's learn about these diagonals in detail.
Face Diagonals of a Cube
In a cube, there are 6 faces and these faces are in the shape of a square. Each face has two diagonals, making it a total of 12 face diagonals in a cube.
Body Diagonal of a Cube
There are 8 vertices in a cube and we know that diagonals join opposite vertices. Hence, there will be four pairs of opposite vertices in a cube. Thus there are four body diagonals in a cube.
Note: There are 16 diagonals in a cube out of which 12 are face diagonals and 4 are body diagonals.
Length of a Diagonal of a Cube
Formula for the Length of a Diagonal of a Cube is given for both faces as well as the body diagonal which is already discussed in the article. Let's learn the formula for each diagonal as follows:
The formula for the face diagonal of a cube is given as
d = √2.a
Where,
- d is the face diagonal of a cube, and
- a is the length of the side of a cube.
We can see that since the face of a cube is of square shape, hence the formula of the face diagonal of a cube is the same as that of the formula of a diagonal of a square.
Formula of Body Diagonal of a Cube
The formula for the Body Diagonal of a Cube is given as
D = √3.a
Where,
- D is the Body Diagonal of a Cube, and
- a is the length of the side of a Cube.
Since there are two types of body diagonals in a cube i.e. face diagonals and body diagonals. Their derivation has been carried out separately under separate headings.
We know that each face of a cube is square-shaped. Let each side of a cube measure 'a' unit. Hence, each side of the square face of a cube measures 'a' unit. In each face, the face diagonal and two adjacent sides of cubes form a right triangle as each angle in a square is 90°. In these right triangles, the diagonal will act as hypotenuse, and the sides will act as perpendicular and base.
Now using Pythagoras Theorem we know that
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
⇒ d2 = a2 + a2
⇒ d = √2.a
Hence, the face diagonal of a cube is √2 times the side of a cube.
Body Diagonal of a Cube Formula Derivation
While deriving the formula for the body diagonal of a cube, the right triangle is formed using a body diagonal, a face diagonal, and a side of a cube. In this right triangle the body diagonal acts as hypotenuse, the face diagonal acts as the base, and the edge(side) of the cube act as the perpendicular.
Let the body Diagonal is represented by D
Side is 'a'
Face diagonal is √2.a {Calculated Above}
By Pythagoras Theorem
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
⇒ D2 = a2 + (√2.a)2
⇒ D2 = a2 + 2a2
⇒ D2 = 3a2
⇒ D = √3.a
Hence, the body diagonal of a Cube measures √3 times the side of the cube.
Read More,
Example 1. Calculate the face diagonal of a cube of side length 2 m.
Solution:
We have,
a = 2
Using the formula we get,
d = √2.a
⇒ d = √2 (2)
⇒ d = (1.414) (2)
⇒ d = 2.83 m
Example 2. Calculate the face diagonal of a cube of side length 5 m.
Solution:
We have,
a = 5
Using the formula we get,
d = √2.a
⇒ d  = √2 (5)
⇒ d = (1.414) (5)
⇒ d = 7.07 m
Example 3. Calculate the body diagonal of a cube of side length 3 m.
Solution:
We have,
a = 3
Using the formula we get,
D = √3.a
⇒ D = √3 (3)
⇒ D = (1.732) (3)
⇒ D = 5.196 m
Example 4. Calculate the body diagonal of a cube of side length 7 m.
Solution:
We have,
a = 7
Using the formula we get,
D = √3.a
⇒ D = √3(7)
⇒ D = (1.732) (7)
⇒ D = 12.12 m
Example 5. Calculate the side length if the face diagonal of a cube is 6 m.
Solution:
We have,
d = 6
Using the formula we get,
d = √2.a
⇒ a = d/√2
⇒ a = 6/√2
⇒ a = 4.24 m
Example 6. Calculate the side length if the body diagonal of a cube is 13√3 m.
Solution:
We have,
D = 13√3
Using the formula we get,
D = √3.a
⇒ a = D/√3
⇒ a = 13√3/√3
⇒ a = 13 m
Example 7. Calculate the body diagonal of a cube if its face diagonal is 9√2 m.
Solution:
We have,
d = 9√2
Find the side length.
d = √2.a
⇒ a = d/√2
⇒ a = 9√2/√2
⇒ a = 9 m
Now using the formula we get,
D = √3.a
⇒ D = √3 (9)
⇒ D = 15.58 m
1: Find the face diagonal of a cube with side 8 cm.
2: Determine the body diagonal of a cube with side 10 cm.
3: Calculate the side length of a cube if its face diagonal is 10 cm.
4: What is the side length of a cube if its body diagonal is 12√3 cm?
5: Find the volume of a cube if its face diagonal is 14√2 cm.
6: A cube has a face diagonal of 5√2 cm. What is its body diagonal?
7: Calculate the surface area of a cube with a body diagonal of 6√3 cm.
8: What is the face diagonal of a cube whose body diagonal is 15 cm?
9: Determine the side length of a cube if the longest pole that can fit inside it is 20 cm.
10: If the face diagonal of a cube is 7 cm, what is its volume?
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