SURFACE AREA OF A CUBOID – DERIVATION AND EXAMPLES

SURFACE AREA OF A CUBOID

What is a cuboid? It is a three-dimensional shape that has six faces and each one of its faces resembles a rectangle. The cuboid has opposite faces equal to each other. The total surface area of a cuboid is the sum of the areas of all its faces. 

A rectangle has length and breadth but a cuboid has length (l), breadth (b) and height (h)as shown in the image below. 

A two-dimensional shape has an area so the area of the rectangle will be length x breadth. A three-dimensional shape has a surface area which is the sum of the area of all the faces of the cuboid. The total surface area of a cuboid is 2 {(l x b) + (b x h) + (h x l)} where l is length, b is breadth and h is height.

For Example: What is the total surface area of a cuboid with the following dimensions?

Ans: The length is 30 cm

         The breadth is 10 cm

And the height is 20 cm so 

Total surface area (TSA) of the cuboid = 2 {(l x b) + (b x h) + (h x l)}

TSA of the cuboid = 2 {(30 x 10) + (10 x 20) + (20 x 30)} = 2200 cm²

The unit of the surface area is written as unit2. ( based on whatever is the value of the unit which can be cm, m etc.)

Now, the top and bottom faces in a cuboid can be labelled as below:

It includes a top and a bottom along with 4 faces. Now, as we saw before, the TSA includes all 6 faces of the cuboid but when we find the lateral surface area (LSA), then we find the area of all the sides excluding the top and the bottom side. 

Lateral Surface area (LSA) of a cuboid is 2h ( l + b) where l is length, b is breadth and h is height.

For example: What is the lateral surface area of the cuboid with length = 30 cm, breadth = 20 cm and height = 15 cm?

Ans: The LSA of a cuboid = 2h (l + b) 

= 2 x 15 (30 + 20)

= 2 x 750

= 1500 cm2

DERIVATION OF THE SURFACE AREA OF A CUBOID

The formulas for the TSA and LSA can be easily derived by understanding that there are 6 faces of a cuboid. 

Now, The total area is simply the sum of the area of all the sides. We will simply find the area of each side as shown below in the table:

Therefore, when we add all the above areas we get 

(l x b) + (l x b) + (b x h) + (b x h) + (h x l) + (h x l) = 2 {(l x b) + (b x h) + (h x l)}

Similarly, we can derive the formula for LSA. We can subtract the area of the top ( ABCD) and the bottom (EFHG) from the TSA to get the area of the four sides. 

So, LSA is

LSA = TSA – {(l x b) + (l x b)} = 2 {(l x b) + (b x h) + (h x l)} – 2 (l x b)

= 2 {(b x h) + (h x l)}

= 2h (l + b)