Question

A cubical box has each edge 8 cm and another cuboidal box is 12 cm long, 8 cm wide and 6 cm high.
(i) Which box has the greater lateral surface area and by how much?
(ii) Which box has the smaller total surface area and by how much?

Answer 1

Hint:Use the formula of lateral surface area of cube and cuboid which is $4{{a}^{2}}$ and $2\left( l+b \right)\times h$ where a is edge length, l is length, b is breadth and h is height. Then use formula for total surface area which is $6{{a}^{2}}$ and $2\left( lb+bh+hl \right)$ for cube and cuboid respectively.

Complete step-by-step answer:
In the question, a cubical box is given of each edge or side given is 8cm and there is another cuboidal box with dimensions given as length equal to 12cm, breadth as 8cm and height as 6cm.
We have to compare the greater lateral surface area and which is greater by how much in the first part of the question. In the second part of the question we have to compare which one has a smaller total surface area and also by how much we have to find it.
As we know that the side length or edge is 8 cm so, we can find its lateral surface area which we can get formula $4{{a}^{2}}$ where ‘a’ represents side length or edge.
So, lateral surface area is,
$4{{a}^{2}}=4\times {{\left( 8 \right)}^{2}}=256c{{m}^{2}}$
Hence, the lateral surface area of the cube is $256c{{m}^{2}}$.
As we know the dimensions of cuboid so, we can find its lateral surface area using formula $2\times \left( l+b \right)\times h$ where l is length, b is breadth and h is height.
So, the lateral surface area of cuboidal box is,
$2\left( l+b \right)\times h=2\left( 12+8 \right)\times 6$
Which is equal to $2\times 20\times 6$ or $240c{{m}^{2}}$.
Hence, the lateral surface area of the cuboid is equal to $240c{{m}^{2}}$.
So, by comparing we can Say that the lateral surface of the cube is greater than that of the cuboidal box by $\left( 256-240 \right)$ or $16c{{m}^{2}}$ .
As we know that side length or edge of the cube. So, we can find its total surface area which we can get by using formula $6{{a}^{2}}$ where ‘a’ represents side length or edge.
So, total surface area is,
$6{{a}^{2}}=6\times {{\left( 8 \right)}^{2}}=396c{{m}^{2}}$
Hence, the total surface area of the cube is $396c{{m}^{2}}$ .
As we know the dimensions of cuboid so, we can find its total surface area using formula $2\times \left( lb+bh+hl \right)$ where l is length, b is breadth and h is height.
So, the total surface area of cuboidal box is,
$2\left( lb+bh+hl \right)$
$=2\left( 12\times 8+8\times 6+6\times 12 \right)$
Which is equal to
$=2\left( 96+48+72 \right)$
$=432c{{m}^{2}}$
So, by comparing we can say that total surface of cube is smaller $\left( 432-396 \right)c{{m}^{2}}$ of $36c{{m}^{2}}$ .
Hence the cube will have greater lateral surface area and smaller total surface area by $16c{{m}^{2}}$ and $36c{{m}^{2}}$ respectively.

Note: Students should be particular and familiar with the formulas of mensuration i.e lateral and total surface area of cube and cuboid as they get confused between lateral and total surface area, so be careful about that.Hence in this question the cube has greater lateral surface area and smaller total surface area by $16c{{m}^{2}}$ and $36c{{m}^{2}}$ respectively.