- Category: Mathematics

**Cube and cuboid** are three-dimensional shapes which consist of three faces and three corners or vertices. Both the shapes looks the same but have different properties. 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**.

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

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.

A Cuboid is made up of six rectangles, each of the rectangle 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.

Any face of a cuboid may be called as 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 rest on is known to be the base of the solid.

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

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.

Hence, AB=CD=GH=EF, AE=DH=BF=CG and EH=FG=AD=BC.

The point of intersection of the 3 edges of a cuboid is called 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 cube has three faces and three 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.

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) cm
^{2} - Area of face BFGC = Area of face AEHD = (b ×h) cm
^{2} - Area of face DHGC = Area of face ABFE = (l ×h) cm
^{2}

**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) |

The sum of surface areas of all sides 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) |

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 |

For cube, length = breadth = height

Suppose length of an edge =l

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

Total Surface Area of Cube= 6l^{2} |

Formula to find Lateral surface area of the cube is:

2(l × l + l × l) = 4l^{2}

LSA of Cube = 4l^{2} |

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 same length. Hence,

Volume of the cube = l^{2} × h

Since, l = h

Therefore,

Volume of the cube = l^{2} × l

Volume of the cube = l^{3} |

- Diagonal of the cuboid =√( l
^{2 }+ b^{2}+h^{2}) - Perimeter of a cuboid = 4 (l + b + h)
- Diagonal of a cube = √3l
- Perimeter of a cube = 12l

**Example 1: Find the total surface area of cuboid with dimensions 2 inches by 3 inches by 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 inches^{2}

** 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 cm^{2}

** 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 = 6a^{2}

= 6× 8^{2} = 6 ×64

= 438 cm^{2}

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.

- 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)

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.