Diagonal of Cuboid
Diagonals of a cuboid are of two types  body diagonal and face diagonal. We can use the diagonal of cuboid formula to calculate the length of the diagonal of cuboid. A cuboid is a threedimensional figure with length, breadth, and height. If all these three measurements are equal, then it is called a cube. We can find the length of the face diagonal using the Pythagoras theorem whereas the length of the body diagonal can be evaluated using the diagonal of cuboid formula. Generally, we refer to the body diagonal as the diagonal of a cuboid.
In this article, we will learn to use the diagonal of cuboid formula to find the length of the face and body diagonal of a cuboid with the help of a few solved examples. We shall derive the diagonal of cuboid formula using the Pythagoras theorem of a rightangled triangle. We will solve some examples based on the concept for a better understanding of the concept.
1.  What is Diagonal of Cuboid? 
2.  Diagonal of Cuboid Formula 
3.  Diagonal of Cuboid Formula Derivation 
4.  Finding Diagonal of Cuboid 
5.  Diagonal of Cuboid and Cube 
6.  FAQs on Diagonal of Cuboid 
What is Diagonal of Cuboid?
The diagonal of cuboid can of two types  body diagonal and the face diagonal. The face diagonal of a cuboid is a lines segment that joins to nonadjacent vertices of the cuboid lying on the same cuboid. On the other hand, the body diagonal of a cuboid is a line segment that joins the two nonadjacent vertices of a cuboid lying in two different faces. Generally, we refer to the body diagonal as the diagonal of cuboid. We can calculate the length of the diagonal using the diagonal of cuboid formula.
Diagonal of Cuboid Definition
The diagonal of cuboid is a line segment that joins two opposite vertices of a cuboid and passes through the body of the cuboid. We have four body diagonals of a cuboid. We have six faces of a cuboid and two diagonals on each face, so there are twelve face diagonals of a cuboid.
Diagonal of Cuboid Formula
If a cuboid has length = l units, width = b units, and height = h units, then we can evaluate the length of the diagonal of a cuboid using the formula √(l^{2} + w^{2} + h^{2}). This formula is used to evaluate the length of the body diagonal. To evaluate the length of the face diagonal of a rectangle, we use the Pythagoras theorem. If a face diagonal lies on a face of a cuboid with dimensions of the face as a × b, then its length is given by, √(a^{2} + b^{2}). The image given below shows the formula for calculating the length of the two diagonals of a cuboid:
Diagonal of Cuboid Formula Derivation
As shown in the cuboid in the previous section, the length of sides of the cuboid is given by,
GH = l units, GF = b units, BF = h units
In right triangle FGH, using the Pythagoras theorem, we have FG^{2} + GH^{2} = HF^{2}
⇒ b^{2} + l^{2} = HF^{2}
⇒ HF = √(l^{2} + b^{2})
So, the length of face diagonal HF is equal to √(l^{2} + b^{2}) units.
Now, in a right triangle HBF, using the Pythagoras theorem, we have
BH^{2} = HF^{2} + BF^{2}
⇒ BH = √(HF^{2} + BF^{2})
= √{[√(l^{2} + b^{2})]^{2} + h^{2}}
= √(l^{2} + b^{2 }+ h^{2})
Therefore, the length of the diagonal of cuboid BH is equal to √(l^{2} + b^{2 }+ h^{2}) units.
Finding Diagonal of Cuboid
Now that we know the diagonal of cuboid formula, we will solve an example and find the diagonal of a cuboid using the formula. Let us consider a cuboid with a length of 8 units, a width of 6 units, and a height of 3 units. Now, finding the length of the face and body diagonal of the cuboid, we have l = 8 units, w = 6 units, h = 3 units. So, the length is given by,
Length of Body Diagonal of Cuboid = √(8^{2} + 6^{2 }+ 3^{2})
= √(64 + 36 + 9)
= √109 units
Hence, the length of the diagonal of cuboid is √109 units.
Diagonal of Cuboid and Cube
Let us now compare the formula for the diagonal of cuboid and cube. A cube is a threedimensional figure with all three dimensions equal. Since, the length, width, and height of a cube are the same, it is easy to find the length of the diagonal of the cube with each side of the length 'a' units. The formulas of cuboid and cube vary as given below:
Formula  Cuboid  Cube 

Face Diagonal  √(l^{2} + b^{2})  √2 a 
Body Diagonal  √(l^{2} + b^{2 }+ h^{2})  √3 a 
Important Notes on Diagonal of Cuboid
 A cuboid has two types of diagonals  face diagonal and body diagonal.
 There are four body diagonals of cuboid and twelve face diagonals.
 A body diagonal is referred to as the diagonal of cuboid.
 The diagonal of cuboid formula is √(l^{2} + b^{2 }+ h^{2}) units.
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Diagonal of Cuboid Examples

Example 1: Find the length of the diagonal of cuboid with dimensions 5 × 3 × 2.
Solution: We have l = 5, b = 3 and h = 2. So, using the diagonal of cuboid formula, the length of the diagonal is given by,
Body diagonal = √(l^{2} + b^{2 }+ h^{2}) units
= √(5^{2} + 3^{2 }+ 2^{2}) units
= √(25 + 9 + 4)
= √(38) units
Answer: The length of the diagonal of the cuboid is equal to √(38) units.

Example 2: Find the length of the face and body diagonal of a cube with each side equal to 7 units.
Solution: We have a = 7
The length of a face diagonal of the cube is given by √2 a = √2 × 7 = 7√2 units
The length of the body diagonal of the cube is equal to √3 a = √3 × 7 = 7√3 units.
Answer: Face diagonal = 7√2 units, Body diagonal = 7√3 units.

Example 3: Find the length of the face diagonal of the cuboid with a length of 10 cm and a width of 5 cm.
Solution: The length of a face diagonal of cuboid with length 10 cm and width 5 cm is given by,
Face diagonal = √(10^{2} + 5^{2})
= √(100 + 25)
= √125
= 5√5 units
Answer: Length of face diagonal = 5√5 units.
FAQs on Diagonal of Cuboid
What is Diagonal of Cuboid in Geometry?
Diagonals of a cuboid are of two types  body diagonal and face diagonal. Generally, we refer to the body diagonal as the diagonal of a cuboid. The diagonal of cuboid is a line segment that joins two opposite vertices of a cuboid that passes through the body of the cuboid.
What is Diagonal of Cuboid Formula?
If a cuboid has length = l units, width = b units, and height = h units, then we can evaluate the length of the diagonal of a cuboid using the formula √(l^{2} + w^{2} + h^{2}). If a face diagonal lies on a face of a cuboid with dimensions of the face as a × b, then its length is given by, √(a^{2} + b^{2}).
How to Find the Length of Diagonal of Cuboid?
We can find the length of diagonal of cuboid using the formula √(l^{2} + w^{2} + h^{2}), where the dimensions of the cuboid are given as length = l units, width = b units, and height = h units.
How Many Diagonals of Cuboid Are There?
We have four body diagonals of cuboid and twelve face diagonals. Each face of the cuboid has two diagonals and there are six faces, so there are twelve face diagonals of a cuboid.
What is the Diagonal of Cuboid and the Cube?
The diagonal of cuboid is a line segment that joins two opposite vertices of a cuboid that passes through the body of the cuboid. And the diagonal of cube is a line segment that joins two opposite vertices of a cuboid that passes through the body of the cube. A cube has all sides equal whereas a cuboid may have different dimensions.
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