A cuboid is a three-dimensional shape that resembles a rectangular box, with 6 faces, 12 edges, and 8 vertices. It is also known as a rectangular prism, characterized by six rectangular faces. A rectangle, a two-dimensional shape, has opposite sides equal and parallel. When congruent rectangles are stacked, a cuboid is formed.
An example of a cuboid in real life is a shoe box. Each face of a cuboid is perpendicular to the adjacent faces.
The shape of cuboid is defined as a closed 3-dimensional figure which is enclosed by rectangular faces. The shape of a cuboid is shown in the figure given below.
Dimensions of a Cuboid
The following are the dimensions of the cuboid:
- Length (L): Generally, the measurement of the longest side of the cuboid, running from one vertex to the opposite vertex is called length, but it is not always true as changing the perspective length can be different than the longer side.
- Width (W): The measurement of the side perpendicular to the length, also running from one vertex to the opposite vertex, is called width.
- Height (H): The measurement of the side perpendicular to both the length and width, running from one vertex to the opposite vertex.
Concepts Related to Cuboid
Every 3D shape has faces, vertices, and edges. There are 6 faces, 8 vertices, and 12 edges in a cuboid. All are shown using the notation given below:
Faces of a Cuboid
There are six rectangular faces in a cuboid. In the figure given above, the six faces are ABFE, DAEH, DCGH, CBFG, ABCD,and EFGH.
The pair of opposite and parallel faces of the given cuboid are given by:
ABCD and EFGH i.e., top and bottom faces respectively
ABFE, DCGH, and DAEH, CBFG are the opposite and parallel faces and are adjacent to the top and bottom faces of the cuboid.
Edges of a Cuboid
Edges of a cuboid are defined as the sides of all the rectangular faces. There are 12 edges in a cuboid which are AB, AD, AE, HD, HE, HG, GF, GC, FE, FB, EF, and CD respectively. The opposite sides of a rectangle are equal and congruent. Hence,
- AB = CD = GH = EF
- AE = DH = BF = CG
- EH = FG = AD = BC
Vertices of a Cuboid
The point at which the 3 edges of a cuboid intersect each other is known as the vertex of a cuboid. A cuboid has 8 vertices.
From the cuboid figure above, the 8 vertices are A, B, C, D, E, F, G, A and H.
Diagonals of a Cuboid
A Cuboid is a 3D shape, so there are two types of diagonals in a cuboid that are,
- Face Diagonal
- Space Diagonal
Below is the Figure demonstrating the diagonals of the Cuboid.
Diagonals of the Cuboid
Face Diagonal
We can draw the face diagonals by connecting the opposite vertices on a particular face of a cuboid. Only two diagonals can be drawn on one face of a cuboid and a cuboid has 6 faces so, a total of 12 face diagonals are there in a cuboid.
Space Diagonal
When we join a line segment from the opposite vertices of a cuboid, we get a space diagonal. The space diagonals transverse through the inner region of the cuboid. Hence, 4 space diagonals can be drawn inside it.
Learn More about Diagonal
Properties of a Cuboid
Following are the properties of a cuboid that help us to understand better:
- There are 3 dimensions in a cuboid i.e., length, width, and height.
- A cube consists of 6 faces, 8 vertices, and 12 edges.
- The angles formed at the vertices of a cuboid are right angles i.e. 90 degrees.
- The opposite edges are parallel to each other.
- All the faces of a cuboid are rectangles in shape.
- Two opposite faces of the cuboid are parallel and congruent to each other.
- Only two diagonals can be drawn on each face of a cuboid.
Some cuboid formulas are added below,
Face Diagonals of Cuboid
The formula for finding the base/top diagonals is given by:
d = √(l2+ w2)
where,
- l is length of a Cuboid
- w is width of a Cuboid
Space Diagonals of Cuboid
The formula for finding the space diagonals is given by:
Space Diagonal of Cuboid = √(l2 + w2 + h2)
where,
- l is length of a Cuboid
- w is width of a Cuboid
- h is height of a Cuboid
Surface Area of Cuboid
Surface Area is defined as the total area occupied by a cuboid shape. A cuboid is a 3D figure so, the surface area will depend on the length, breadth, and height.
There are two types of surface areas
- Total surface area
- Lateral surface area
Total Surface Area of Cuboid
Thelateral Surface Area of Cuboid includes the area of all the faces of a cuboid. So, the formula for Total Surface Area is given as:
TSA = 2(lb + bh + lh) square units
Lateral Surface Area of Cuboid
The except Surface Area of a Cuboid includes the area of the faces of a cuboid except the base and the top. So, the formula for Lateral Surface Area is given as:
LSA = 2h(l + b) square units
where,
- l is the Length of Cuboid,
- b is the Breadth of Cuboid, and
- h is the Height of Cuboid.
Read more about: Surface Area of Cuboid
Volume of Cuboid
The volume of a cuboid is defined as the space occupied by a cuboid. The volume of a cuboid depends on its length, breadth, and height. Thus, modifying any one of these quantities modifies the volume of the shape.
Cubic units are the unit of the cuboid's volume. Hence, the formula to calculate the volume of a cuboid is given by:
Volume of Cuboid = Base Area × Height
V = l × b × h
where,
- l is the Length of Cuboid
- b is the Breadth of Cuboid
- h is the Height of Cuboid
- V is Volume of a Cuboid
Read More, Volume of Cuboid,
Net of Cuboid
When we open a 3D shape, we get its net. So, the net of a cuboid can be referred to when a 3D shape opens into a flat object making it into a 2D shape.
The net of cuboid shape helps to understand the sides that are rectangular in shape in a better way. Once the flattened 2D shape is folded back together, the shape of a cuboid is again formed. The image of net of cube is added below,
Cube vs Cuboid
The key differences between both cubes and cuboids are listed in the following table:
Difference between Cube and Cuboid |
|---|
| Characteristic | Cube | Cuboid |
|---|
| Definition | A cube is a three-dimensional shape with all sides of equal length. | A cuboid is a three-dimensional shape with six rectangular faces, and all angles are right angles. |
|---|
| Shape | All sides are equal in length, and all angles are right angles. | Opposite sides are equal in length, and all angles are right angles. |
|---|
| Symmetry | Highly symmetrical with equal sides and angles. | Less symmetrical compared to a cube. |
|---|
| Faces | 6 square faces. | 6 rectangular faces. |
|---|
| Volume Formula | V = s3 | V = l × w × h |
|---|
| Surface Area Formula | A = 6s2 | A = 2(lw + lh + wh) |
|---|
| Examples | A standard die or a Rubik's Cube. | A rectangular box, a book, or a room. |
|---|
Read More,
Examples on Cuboid
Example 1: Find the height of a cuboid given that its total surface area is 108 sq. units, length 4 units, and breadth 6 units.
Solution:
Let the height be h units
Given,
- Total Surface Area(a) = 108 sq. units
- Length(l) = 4 units
- Breadth(b) = 6 units
We know, h = (a - 2.lb)/{2(l + b)}
⇒ h = {108 - 2.(4)(6)}/{2.(4 + 6)} = 60/20
⇒ h = 3 units
Example 2: Determine the lateral surface area of a cuboid if its length, breadth, and height are 15 in, 8 in, and 12 in, respectively.
Solution:
Given,
- Length of a cuboid (l) = 15 in
- Breadth of a cuboid (b) = 8 in
- Height of a cuboid (h) = 12 in
We have,
Lateral Surface Area of a Cuboid(A) = 2h(l + b)
A = 2 × 12 (15 + 8)
A = 24 × 23 = 552 square inches.
Hence, lateral surface area of the given cuboid is 552 square inches.
Example 3: Robert has to cover the edges of a rectangular box with a tape. How much minimum tape does he require if the dimensions of the cuboid are 16 in × 10 in × 8 in?
Solution:
Since Robert has to cover the edges of a box, so he has to find the perimeter of the box i.e., cuboid.
Given,
- Length of a cuboid (l) = 16 in
- Breadth of a cuboid (b) = 10 in
- Height of a cuboid (h) = 8 in
Perimeter of Cuboid(P) = 4(l + w + h)
P = 4(16 + 10 + 8) = 4 × 34 = 136 inches
Hence, the minimum tape required to cover the edges of a rectangular box with a tape is 136 inches.
Example 4: If the length and width of a cuboid are 6 inches and 8 inches respectively, what will be the value of the areface diagonal?
Solution:
Given,
- Length of a cuboid (l) = 6 in
- Width of a cuboid (w) = 8 in
Face Diagonal(d) = √(l2 + w2)
d = √(62 + 82) = √(100) = 10 inches
Hence, the value of face diagonal of the cis 10 inches.
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