 # Cube and Cuboid

Cube and cuboid are three-dimensional shapes which consist of six faces, eight vertices and twelve edges. The primary difference between them is a cube has all its sides equal whereas the length, width and height of a cuboid are different. Both shapes look almost the same but have different properties. The area and volume of cube, cuboid and also cylinder differ from each other.

In everyday life, objects like a wooden box, a matchbox, a tea packet, a chalk box, a dice, a book, etc are encountered. All these objects have a similar shape. In fact, all these objects are made of six rectangular planes. The shape of these objects is either a cuboid or cube. Here, in this article, we will learn the difference between the two shapes with the help of their properties and formulas of surface area and volume.

## Definition of Cube and Cuboid

Cube: A cube is a three-dimensional shape which is defined XYZ plane. It has six faces, eight vertices and twelve edges. All the faces of the cube are in square shape and have equal dimensions.

Cuboid: A cuboid is also a polyhedron having six faces, eight vertices and twelve edges. The faces of the cuboid are parallel. But not all the faces of a cuboid are equal in dimensions.

### Difference Between Cube and Cuboid

• The sides of the cube are equal but for cuboid they are different.
• The sides of the cube are square in shape but for cuboid, they are in a rectangular shape.
• All the diagonals of the cube are equal but a cuboid has equal diagonals for only parallel sides.

## Shape of Cube and Cuboid

As we already know both cube and cuboid are in 3D shape, whose axes goes along the x-axis, y-axis and z-axis plane. Now let us learn in detail.

A cuboid is a closed 3-dimensional geometrical figure bounded by six rectangular plane regions. ## Properties of a Cuboid

Below are the properties of cuboid, its faces, base and lateral faces, edges and vertices.

### Faces of Cuboid

A Cuboid is made up of six rectangles, each of the rectangles is called the face. In the figure above, ABFE, DAEH, DCGH, CBFG, ABCD and EFGH are the 6 faces of cuboid.

The top face ABCD and bottom face EFGH form a pair of opposite faces. Similarly, ABFE, DCGH, and DAEH, CBFG are pairs of opposite faces. Any two faces other than the opposite faces are called adjacent faces.

Consider a face ABCD, the adjacent face to this are ABFE, BCGF, CDHG, and ADHE.

### Base and lateral faces

Any face of a cuboid may be called the base of the cuboid. The four faces which are adjacent to the base are called the lateral faces of the cuboid. Usually, the surface on which a solid rests on is known to be the base of the solid.

In Figure (1) above, EFGH represents the base of a cuboid.

### Edges

The edge of the cuboid is a line segment between any two adjacent vertices.

There are 12 edges, they are AB, AD, AE, HD, HE, HG, GF, GC, FE, FB, EF and CD and the opposite sides of a rectangle are equal.

### Vertices of Cuboid

The point of intersection of the 3 edges of a cuboid is called the vertex of a cuboid.

A cuboid has 8 vertices A, B, C, D, E, F, G and H represents vertices of the cuboid in fig 1.

By observation, the twelve edges of a cuboid can be grouped into three groups, such that all edges in one group are equal in length, so there are three distinct groups and the groups are named as length, breadth and height.

A solid having its length, breadth, height all to be equal in measurement is called a cube. A cube is a solid bounded by six square plane regions, where the side of the cube is called edge.

## Properties of Cube

• A cube has six faces and twelve edges of equal length.
• It has square-shaped faces.
• The angles of the cube in the plane are at a right angle.
• Each face of the cube meets four other faces.
• Each vertex of the cube meets three faces and three edges.
• Opposite edges of the cube are parallel to each other.

## Cube and Cuboid Formulas

The formulas for cube and cuboid are defined based on their surface areas, lateral surface areas and volume.

 Cube Cuboid Total Surface Area = 6(side)2 Total Surface area = 2 (Length x Breadth + breadth x height + Length x height) Lateral Surface Area = 4 (Side)2 Lateral Surface area = 2 height(length + breadth) Volume of cube = (Side)3 Volume of the cuboid = (length × breadth × height) Diagonal of a cube = √3l Diagonal of the cuboid =√( l2 + b2 +h2) Perimeter of cube = 12 x side Perimeter of cuboid = 4 (length + breadth + height)

### Surface Area of Cube and Cuboid

The surface area of a cuboid is equal to the sum of the areas of its six rectangular faces. Consider a cuboid having the length to be ‘l’ cm, breadth be ‘b’ cm and height be ‘h’ cm.

• Area of face EFGH = Area of Face ABCD = (l× b) cm2
• Area of face BFGC = Area of face AEHD = (b ×h) cm2
• Area of face DHGC = Area of face ABFE = (l ×h) cm2

Total surface area of a cuboid = Sum of the areas of all its 6 rectangular faces

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

Example: If the length, breadth and height of a cuboid are 5 cm, 3 cm and 4 cm, then find its total surface area.

Given, Length, l = 5 cm, Breadth, b = 3 cm and Height, h = 4 cm.

TSA = 2(lb+bh+lh)

TSA = 2(5 x 3 + 3 x 4 + 5 x 4)

TSA = 2(15 + 12 + 20)

TSA = 2(47) = 94 sq.cm.

Lateral surface area of a Cuboid:

The sum of surface areas of all faces except the top and bottom face of solid is defined as the lateral surface area of a solid.

Consider a Cuboid of length, breadth and height to be l, b and h respectively.

Lateral surface area of the cuboid= Area of face ADHE + Area of face BCGF + Area of face ABFE + Area of face DCGH

=2(b × h) + 2(l × h)

=2h(l + b)

 LSA of Cuboid = 2h(l +b)

 Example: If the length, breadth and height of a cuboid are 5 cm, 3 cm and 4 cm, then find its lateral surface area. Given, Length = 5 cm, Breadth = 3 cm and Height = 4 cm LSA = 2h(l+b) LSA = 2 x 4(5+3) LSA = 2 x 4(8) LSA = 2 x 32 = 64 cm2

Surface Area of a Cube:

For cube, length = breadth = height

Suppose the length of an edge = l

Hence, surface area of the cube = 2(l × l +l × l + l × l) = 2 x 3l = 6l2

 Total Surface Area of Cube= 6l2

 Example: If the length of the side of the cube is 6 cm, then find its total surface area. Given, side length = 6 cm TSA of cube = 6l2  TSA = 6 (6)2 TSA = 6 x 36 TSA = 216 sq.cm

Lateral surface area of a Cube:

Formula to find Lateral surface area of the cube is:
2(l × l + l × l) = 4l2

 LSA of Cube = 4l2

 Example: If the length of the side of the cube is 6 cm, then find its lateral surface area. Given, Side length, l = 6 cm LSA of cube = 4l2  LSA = 4 (6)2 LSA = 4 x 36 = 144 sq.cm

### Volume of the Cube and Cuboid

Volume of Cuboid:

The volume of the cuboid is equal to the product of the area of one surface and height.

Volume of the cuboid = (length × breadth × height) cubic units

 Volume of the cuboid = ( l × b × h) cubic units

 Example: If the length, breadth and height of a cuboid are 5 cm, 3 cm and 4 cm, then find its lateral surface area. Given, Length (l) = 5 cm, Breadth (b) = 3 cm and Height (h) = 4 cm Volume of cuboid = l x b x h V = 5 x 3 x 4 V = 60 cubic cm

Volume of the Cube:

The volume of the cube is equal to the product of the area of the cube and height. As we know already, all the edges of the cube are of the same length. Hence,

Volume of the cube = l2 × h

Since, l = h

Therefore,

Volume of the cube = l2 × l

 Volume of the cube = l3

 Example: If the length of the side of the cube is 6 cm, then find its lateral surface area. Given, side length = 6 cm Volume of cube = side3 = 63 V = 216 cubic.cm

### Diagonal of Cube and Cuboid

The length of diagonal of the cuboid is given by:
Diagonal of the cuboid =√( l+ b2 +h2)

The length of diagonal of a cube is given by:

Diagonal of a cube = √3l

### Perimeter of Cube and Cuboid

The perimeter of the cuboid is based on its length, width and height. Since the cuboid has 12 edges and the value of its edges are different from each other, therefore, the perimeter is given by:

Perimeter of a cuboid =  4 (l + b + h)

where l is the length

h is the height

 Example: If the length, width and height of a cuboid are 5 cm, 3 cm and 4 cm, find its Perimeter. Given, Length = 5 cm, Width = 3 cm and Height = 4 cm Perimeter = 4 (l + b + h) = 4 (5+3+4) P = 4 (12) P = 48 cm

The perimeter of the cube also depends upon the number of edges it has and the length of the edges. Since the cube has 12 edges and all the edges have equal length, therefore the perimeter of the cube is given by:

Perimeter of a cube =  12l

where l is the length of the edge of the cube.

 Example: If the side length of the cube is 6 cm, then find its perimeter. Given , l = 6cm The perimeter of cube = 12l  P = 12 x 6 P = 72 cm

## Example Problems

Example 1: Find the total surface area of the cuboid with dimensions 2 inches × 3 inches × 7 inches.

Solution: Total Surface Area(TSA) = 2 (lb + bh + hl )

TSA = 2 ( 2×3 + 3×7 + 7×2)

TSA = 2 ( 6 + 21 + 14 )

TSA = 82

So, the total surface area of this cuboid is 82 inches2

Example 2:  The length, width and height of a cuboid are 12 cm, 13 cm and 15 cm, respectively. Find the lateral surface area of a cuboid.

Solution: Lateral surface area of a cuboid is given by:

LSA = 2h ( l + w )

LSA = 2×15 ( 12 + 13 )

LSA = 750 cm2

Example 3: Find the surface area of a cube having its sides equal to 8 cm in length.

Solution: Given length, ‘a’= 8 cm

Surface area = 6a2

= 6× 82

= 6 ×64

= 384 cm2

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## Video Lesson ### What are a cube and a cuboid?

A cube is a three-dimensional shape having all its sides equal and the faces of the cube are square in shape.
A cuboid is also a three-dimensional shape which has three pair of equal sides parallel to each other and the faces of the cuboid are all in a rectangular shape.
How many edges does a cuboid have?

### How many edges does a cuboid have?

A cuboid has 12 edges, six faces and eight vertices.

### What the difference between cuboid and cube?

A cube is a three-dimensional figure whose all sides are equal i.e. all of its 6 faces are square. On the contrary, a cuboid is a three-dimensional figure whose all sides are not equal and all of its 6 faces are rectangles.

### What are the formulas of cube and cuboid?

• Total surface area:

Cube= 6× (side)2

Cuboid= 2(lb + bh +lh)

• Lateral surface area:

Cube= 4× (side)2

Cuboid= 2h(l +b)

• Volume:

Cube= (side)3

Cuboid= (length × breadth × height)

### Is a cube, a special kind of cuboid?

Yes, a cube is a special kind of cuboid where all the faces of the cuboid are of equal length. In a cuboid, there are 6 faces which are rectangles. If the rectangles have equal sides, they become squares and eventually, the cuboid becomes a cube.