Cube & Cuboid - Total Surface Area & Lateral Surface Area of a Cube & a Cuboid

 

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 a cuboid.

 

Cuboid:

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

Cuboid

Figure 1

Face – 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.

Base and lateral faces: 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 cube rest on is known to be the base of the cube.

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.

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

Vertex – 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 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.

 

Cube:

A cuboid having its length, breadth, height all to be equal in measurement is called as a cube.

A cube is a solid bounded by six square plane regions, where the side of the cube is called edge.

Total Surface area :

Cuboid –

Surface area of a cuboid

The surface area of a cuboid is equal to the sum of the areas of its six rectangular faces.

Consider a cuboid having 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

= 2(lb + bh +lh)

Cube –

Total Surface area of a cube:

For cube, length = breadth = height

Suppose length of an edge =l

Hence, surface area of the cube = 2(l × l +l × l + l × l)

=\(2\times (3l^{2})=6l^{2}\)

 

Lateral surface area :

Cuboid –

The sum of surface areas of all sides except top and bottom face of solid is defined as 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 cuboid

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)

Cube –

Lateral surface area of cube

\(2(l × l + l × l) ~= ~4l^2\)

Note:

  • Diagonal of the cuboid</> = \(\sqrt{l^2~+~b^2~+~h^2}\)
  • Perimeter of a cuboid</> = \( 4 (l + b + h)\)
  • Diagonal of a cube</> = \(\sqrt{3}l\)
  • Perimeter of a cube</> = \( 12l\)

 

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

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 \( = 6a^{2}\)

\(= 6\times 8^{2} = 6 \times 64\)

\(= 438 \; cm^{2}\)

To learn more math concepts in a simplest and coolest way, download Byju’s – The Learning App from Google Play Store. Also, visit our page to know more about geometrical shapes and to learn about Surface areas and volumes of different figures.


Practise This Question

The cost of painting the inner curved surface area of a cylindrical container of depth 20 m is Rs. 8800.  If painter takes Rs. 40 per metre square to paint the cylindrical container, then find radius of the cylindrical vessel.
Take π=227