Question

A cubical box has each edge $10cm$ and another cuboidal box is $12.5cm$ long, $10cm$ wide and $8cm$ 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

(i) Edge of cube = 10 cm

Length (l) of box = 12.5 cm

Breadth (b) of box = 10 cm

Height (h) of box = 8 cm

Lateral surface area of cubical box $=4{\text{(edge)}}^2$

$=4(10cm{)}^2$

$=400c{m}^2$

Lateral surface area of cuboidal box $=2[lh+bh]$

$=\left[2\right(12.5\times 8+10\times 8\left)\right]c{m}^2$

$=2(100+80)c{m}^2$

$=(2\times 180)c{m}^2$

$=360c{m}^2$

Clearly, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box.

Lateral surface area of cubical box $-$ Lateral surface area of cuboidal box $=400c{m}^2-360c{m}^2=40c{m}^2$

Therefore, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box by $40c{m}^2$.

(ii) Total surface area of cubical box $=6{\text{(edge)}}^2=6\left(100\right)c{m}^2=600c{m}^2$

Total surface area of cuboidal box$=2[lh+bh+lb]$

$=\left[2\right(12.5\times 8+10\times 8+12.5\times 10]c{m}^2$

$=610c{m}^2$

Clearly, the total surface area of the cubical box is smaller than that of the cuboidal box.

Total surface area of cuboidal box − Total surface area of cubical box $=610c{m}^2-600c{m}^2=10c{m}^2$

Therefore, the total surface area of the cubical box is smaller than that of the cuboidal box by $10c{m}^2$.