Cuboid

In geometry, a convex polyhedron that is bounded by six rectangular faces with eight vertices and twelve edges is called a cuboid. It is a three dimensional shape whose axes are x, y and z. In maths, we can observe other shapes which are exactly the same as cuboid, they are rectangular cuboid, rectangular box, right rectangular prism, right cuboid, rectangular parallelepiped, and rectangular hexahedron.

Table of contents:

Cuboid Shape

The cuboid shape is a closed 3d figure that is enclosed by rectangular faces that means plane regions of rectangles. The shape of a cuboid is shown in the below figure.

Cuboid shape

Cuboid Faces Edges Vertices

A cuboid has 6 faces, 8 vertices and 12 edges. All these can be shown using notation as given below:

Faces: A Cuboid is made up of six rectangular faces. In the above given figure, the six faces are:

ABFE

DAEH

DCGH

CBFG

ABCD

EFGH

The pair of opposite and parallel faces of the given cuboid are:

ABCD and EFGH (top and bottom faces respectively)

ABFE, DCGH, and DAEH, CBFG (opposite and parallel faces which are adjacent to top and bottom faces of the cuboid)

For each face, we can write the remaining faces as its adjacent faces such as the face ABCD is the adjacent face to ABFE, BCGF, CDHG, and ADHE.

Edges: The sides of all the rectangular faces are referred to as the edges of the cuboid. As we know, there are 12 edges for a cuboid. They are AB, AD, AE, HD, HE, HG, GF, GC, FE, FB, EF and CD respectively. Also, the opposite sides of a rectangle are equal. Therefore,

AB = CD = GH = EF

AE = DH = BF = CG

EH = FG = AD = BC

Vertices: The point of intersection of the 3 edges of a cuboid is called the vertex of a cuboid and a cuboid has 8 vertices.

From the given cuboid figure, the 8 vertices are A, B, C, D, E, F, G and H.

Let us have a look at the visualization of faces, edges and vertices of a cuboid.

Cuboid faces, edges and vertices

Cuboid Surface Area

The surface area of any three-dimensional shape is the total region covered by all its faces. In the same way, the surface area of a cuboid is the sum of areas of all its six rectangular faces. The surface area of cuboids can be divided into two types namely lateral surface area and total surface area. The formulas for these can be derived from the figure given below.

Cuboid dimensions

Let l, b and h be the length, breadth and height of a cuboid respectively.

From the figure given above,

AB = CD = EF = GH = l

BC = AD = EH = FG = b

CG = BF = AE = DH = h

Cuboid Lateral Surface Area

Lateral surface area (LSA) of a cuboid is the sum of areas of all faces except the top and bottom faces.

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

= (AD × DH) + (BC × CG) + (AB × BF) + (CD × BC)

= (b × h) + (b × h) + (l × h) + (l × h)

= 2(b × h) + 2(l × h)

= 2h(l + b)

Lateral Surface Area (LSA) = 2h(l + b) sq.units

Cuboid Total Surface Area

Total surface area (TSA) of a cuboid is the sum of areas of all its rectangular faces.

Area of face EFGH = Area of Face ABCD = (l × b)

Area of face BFGC = Area of face AEHD = (b × h)

Area of face DHGC = Area of face ABFE = (l × h)

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

= Area of (ABCD + EFGH + BFGC + AEHD + DHGC + ABFE)

= (l × b) + (l × b) + (b × h) + (b × h) + (l × h) + (l × h)

= 2lb + 2bh + 2hl

= 2(lb + bh + hl)

Total Surface Area (TSA) = 2(lb + bh + hl) sq.units

Cuboid Volume

Volume of a solid is equal to the product of base area and its height. Thus, the volume of cuboid is equal to the product of base rectangular face and height.

Volume = (Length × Breadth) × Height

= (l × b) × h

= lbh

Volume (V) = (l × b × h) cubic units

Cuboid Diagonal

The length of the diagonal of a cuboid of dimensions l, b and h is given by the formula:

Diagonal = √(l2 + b2 + h2)

Diagonal = √(l2 + b2 + h2) units

Cuboid Perimeter

The perimeter of a cuboid will be the sum of lengths of all the edges. Thus, from the above figure 2,

AB = CD = EF = GH = l (length)

BC = AD = EH = FG = b (breadth)

CG = BF = AE = DH = h (height)

Perimeter of the cuboid = AB + CD + EF + GH + BC + AD + EH + FG + CG + BF + AE + DH

= (l + l + l + l) + (b + b + b + b) + (h + h + h + h)

= 4l + 4b + 4h

= 4(l + b + h)

Perimeter (P) = 4(l + b + h) units

Cuboid Formulas

The table below shows the formulas of a cuboid of length (l), breadth (b) and height (h).

Measure

Formula

Lateral Surface Area (LSA)

2h(l + b)

Total Surface Area (TSA)

2(lb + bh + hl)

Volume

lbh

Diagonal

√(l2 + b2 + h2)

Perimeter

4(l + b + h)

Cuboid Net

The net of a solid shape is the plane obtained by unfolding it across a line. This net will again form the original solid when we fold it. The number of different nets for a cuboid or rectangular cuboid with 3 different lengths are 54.

Click here to learn more about nets of 3d shapes.

Net 1:

Cuboid net 1

Net 2:

Cuboid net 2

Similarly, the nets of a cuboid can be shown in different ways.

Cuboid Example

Below are some example problems solved using the formulas of cuboid.

Example 1:

Find the volume of a cuboid of length 10 cm, breadth 8 cm and height 4 cm.

Solution:

Given,

Length = l = 10 cm

Breadth = b = 8 cm

Height = h = 4 cm

Volume = lbh

= 10 × 8 × 4

= 320 cm3

Example 2:

Calculate the lateral and total surface area of a cuboid of dimensions 12 cm × 7 cm × 5 cm.

Solution:

Given dimensions of a cuboid are 12 cm × 7 cm × 5 cm.

i.e. l = 12 cm, b = 7 cm, h = 5 cm

Lateral surface area (LSA) = 2h(l + b)

= 2 × 5 (12 + 7)

= 10 × 19

= 190 cm2

Total surface area (TSA) = 2(lb + bh + hl)

= 2(12 × 7 + 7 × 5 + 5 × 12)

= 2(84 + 35 + 60)

= 358 cm2

Frequently Asked Questions on Cuboid

What is the difference between cube and cuboid?

The main difference between the cube and cuboid is their faces, i.e. the cube has six square shaped faces, whereas cuboid has six rectangular faces. That means, all the edges of the cube are equal and the edges of a cuboid can be written as 3 groups of equal edges.

What is the cuboid formula?

The formulas of cuboid are:

Lateral surface area (LSA) = 2h(l + b)

Total surface area = TSA = 2(lb + bh + hl)

Volume (V) = lbh

Perimeter (P) = 4(l + b + h)

Diagonal (D) = √(l^2 + b^2 + h^2)

What is the difference between cuboid and rectangular prism?

There is no difference between cuboid and rectangular prism since both will have 6 rectangular faces, 8 vertices and 12 edges. Also, both look the same as a box.

Can a cuboid have a square face?

As we know, a cube is the special kind of cuboid with all its edges equal in length. Hence, cuboid can have a square face.

How do you find the area of a cuboid?

To find the total surface area of a cuboid, add the areas of all the six faces. Suppose l, b and h be the length, breadth and height of a cuboid, then the total surface area will be 2(lb + bh + hl).

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