Cuboid – Shape and Properties

Cuboid is a three-dimensional shape that looks like a rectangular box in our everyday life. Cuboids have 6 faces, 12 edges, and 8 vertices. A cuboid is also called a rectangular prism. Example of a cuboid in real life is a shoe box.

In this article, we will learn about all things cuboid such as definition, shape, dimensions, and others in detail.

What is Cuboid?

Cuboid, also known as a rectangular prism, is a three-dimensional geometric shape characterized by six rectangular faces. As we know, a rectangle is defined as a two-dimensional flat shape having opposite sides equal and parallel. Now, what if we place congruent rectangles on top of each other? We will get the three-dimensional shape cuboid.

Cuboid Definition

Cuboid is a three dimensional solid which has 6 faces, 12 edges and 8 vertices and where each face is perpendicular to the adjacent faces.

Shape of a Cuboid

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.

Faces, Edges and Vertices of a 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 notation as given below:

Cuboid follows the Euler’s Formula and the relation between Faces (F), Vertices (V) and Edges (E) of a cuboid satisfies the Euler’s Equation: F + V = E + 2

6 + 8 = 12 + 2

14 = 14

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 which 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 and H.

Diagonals of a Cuboid

Cuboid is a 3D shape, so there are two types of diagonals in a cuboid that are,

• Face Diagonal
• Space Diagonal

Let’s learn about them in detail.

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 which helps 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 rectangle 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.

Cuboid Formula

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

Total 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

Lateral Surface Area of a Cuboid includes the area of the faces of a cuboid expect 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 cuboid depends on its length, breadth, and height. Thus, modifying any one of these quantities modifies the volume of the shape.

Cubic units is 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

What is Cuboid Formula?

The following table provide all the formulas related to cuboid:

Formulas of Cuboid
Property	Formula
Face Diagonal	√(l2 + w2)
Space Diagonal	√(l2 + w2 + h2)
Perimeter	4(l + w + h)
Total Surface Area	2(lb + bh + lh) Square Units
Lateral Surface Area	2h(l + b) Square Units
Volume	l × b × h Cubic Units

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 cube 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,

• Mensuration
• Surface Area and Volume
• Faces, Vertices and Edges

Sample Questions on Cuboid

Some questions base on Cuboid are,

1. How Many Faces does a Cuboid have?

A cuboid has six faces.

2. How Many Edges Does a Cuboid Have?

A cuboid has 12 edges.

3. How Many Vertices Does a Cuboid Have?

A cuboid has 8 vertices (corners).

4. How to Get the Volume of a Cuboid?

To find the volume of a cuboid, use the formula: Volume = length × width × height.

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 face 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.

Example 5: Determine the length and the total surface area of a cuboid whose lateral surface area is 960 sq. in and whose breadth and height are 12 in and 16 in, respectively.

Solution:

Given,

• Breadth of a Cuboid (b) = 12 in
• Height of a Cuboid (h) = 16 in
• Lateral Surface Area = 960 square inches

We know that,

Lateral Surface Area of Cuboid(LSA) = 2h(l + b)

⇒ 2 × 16 (l + 12) = 960

⇒ 32 (l + 12) = 960

⇒ (l + 12) = 960/32 = 30

⇒ l = 30 – 12 = 18 in

We have,

Total Surface Area of Cuboid(TSA) = 2 (lb + bh + lh)

TSA = 2 [(18 × 12) + (12 × 16) + (18 × 16)] = 2 [ 216 + 192 + 288]

TSA = 2 × [696] = 1398 square inches

Hence, length and total surface area of cuboid are 18 in and 1398 sq. in, respectively.

Practice Problems on Cuboid

Various practice problems on Cuboid are added below,

Problem 1: How many edges, vertices and faces are there in a cuboid?

Problem 2: Calculate the height of the cuboid whose lateral surface area is 360 square units and whose length and breadth are 12 units and 8 units, respectively.

Problem 3: Calculate the total surface area of a cuboid if its length, breadth, and height are 10 in, 5 in, and 8 in, respectively.

Problem 4: What is the value of space diagonal if length, breadth, and height are 13 in, 10 in, and 12 in, respectively.

Problem 5: Find the cost of painting the walls of a room if the length, breadth and height of the room are 24 feet, 18 feet and 10 feet respectively and the cost of painting the wall is Rs 20 per square feet.

Frequently Asked Questions on Cuboid

What is a Cuboid?

A cuboid is a three-dimensional geometric shape with six rectangular faces, 12 edges, and 8 vertices. Cuboids are also called rectangular prisms.

What are Properties of a Cuboid?

Some Properties of a Cuboid are:

• Six rectangular faces.
• Opposite faces are congruent and parallel.
• 12 edges, where each edge is shared by two faces.
• 8 vertices or corners.
• Diagonals connecting opposite vertices.

How Is a Cuboid Different from a Cube?

A cube is a special type of cuboid with all sides of equal length, making it a regular polyhedron. In contrast, a cuboid has unequal dimensions, allowing for rectangular faces.

What is Formulas for Surface Area of a Cuboid?

Surface Area (SA) of a cuboid can be calculated using the formula: SA = 2lw + 2lh + 2wh

What is the Formulas for the Volume of a Cuboid?

Volume (V) of a cuboid can be calculated using the formula: V = l × w × h

What Are Examples of Cuboids?

Some Examples of Cuboids in real life includes,

• Most books have a cuboidal shape.
• Shipping boxes and rectangular containers.
• Building blocks, like bricks and cinder blocks.

Can a Cuboid have Unequal Angles?

A cuboid’s angles are always right angles (90°). It does not have unequal angles.

What is Perimeter of Cuboid?

Perimeter is only calculated for a 2-D object. Since cuboid is a 3-D object is perimeter is not calculated. But we can find the sum of all the sides of cuboid and term it as perimeter of cuboid. Perimeter of Cuboid(P) = 4(l + b + h) units.

What is Cube and Cuboid?

Cube and Cuboid are 3-dl shapes with six faces, eight vertices and twelve edges. The basicy difference between them is that a cube has all its sides equal whereas in a cuboid length, breadth, and height are different.