Question

Visualise the area of a cuboid.

Answer 1

Step 1: Drawing the figure

Let $\mathrm{l},\mathrm{b}$ and $\mathrm{h}$ be the length, breadth, and height of a cuboid respectively.

According to the figure, we have,

$$\begin{gathered} \mathrm{AB}=\mathrm{CD}=\mathrm{EF}=\mathrm{GH}=\mathrm{l} \\ \mathrm{BC}=\mathrm{AD}=\mathrm{EH}=\mathrm{FG}=\mathrm{b} \\ \mathrm{CG}=\mathrm{BF}=\mathrm{AE}=\mathrm{DH}=\mathrm{h} \end{gathered}$$

Step 2: Finding the curved surface area

The curved surface area (CSA) of a cuboid is the sum of areas of all its rectangular faces excluding the top and the bottom face i.e. area of four faces except top and bottom face.$\begin{array}{rcl}\mathrm{Lateral}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cuboid}& =& \mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{ADHE}+\mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{BCGF}+\mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{ABFE}+\mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{DCGH}\\ & =& (\mathrm{AD}\times \mathrm{DH})+(\mathrm{BC}\times \mathrm{CG})+(\mathrm{AB}\times \mathrm{BF})+(\mathrm{CD}\times \mathrm{BC})\\ & =& (\mathrm{b}\times \mathrm{h})+(\mathrm{b}\times \mathrm{h})+(\mathrm{l}\times \mathrm{h})+(\mathrm{l}\times \mathrm{h})\\ & =& 2(\mathrm{b}\times \mathrm{h})+2(\mathrm{l}\times \mathrm{h})\end{array}$

Hence, Lateral surface area of the cuboid$=2\mathrm{h}(\mathrm{l}+\mathrm{b})\mathrm{sq}.\mathrm{units}$

Step 3: Finding the total surface area

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

Area of face $\mathrm{EFGH}=\mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{ABCD}=(\mathrm{l}\times \mathrm{b})$

Area of face $\mathrm{BFGC}=\mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{AEHD}=(\mathrm{b}\times \mathrm{h})$

Area of face $\mathrm{DHGC}=\mathrm{Area}\mathrm{of}\mathrm{face}\mathrm{ABFE}=(\mathrm{l}\times \mathrm{h})$

$\begin{array}{rcl}\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{a}\mathrm{cuboid}& =& \mathrm{Sum}\mathrm{of}\mathrm{the}\mathrm{areas}\mathrm{of}\mathrm{all}\mathrm{its}6\mathrm{rectangular}\mathrm{faces}\\ & =& \mathrm{Area}\mathrm{of}(\mathrm{ABCD}+\mathrm{EFGH}+\mathrm{BFGC}+\mathrm{AEHD}+\mathrm{DHGC}+\mathrm{ABFE})\\ & =& (\mathrm{l}\times \mathrm{b})+(\mathrm{l}\times \mathrm{b})+(\mathrm{b}\times \mathrm{h})+(\mathrm{b}\times \mathrm{h})+(\mathrm{l}\times \mathrm{h})+(\mathrm{l}\times \mathrm{h})\\ & =& 2\mathrm{lb}+2\mathrm{bh}+2\mathrm{hl}\end{array}$

Hence, Total surface area of a cuboid$=2(\mathrm{lb}+\mathrm{bh}+\mathrm{hl})\mathrm{sq}.\mathrm{units}$