# Volume of a cuboid

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Think of a stack of sheets, in the shape of a cuboid. The dimensions of each sheet in the stack are $$l \times b$$ while the total height of the stack is $$h$$. How much volume does this stack occupy? The volume will depend on two factors:

• the dimensions of the sheets. The larger the area A of the sheets, the larger will be the volume.

• the height $$h$$ of the stack: the larger the height, the larger the volume

Thus, we can say that the volume of the cuboid is

$V = A \times h = l \times b \times h$

For example, for a cuboid of dimensions 5 cm × 4 cm × 3 cm, its volume is

$V = {\rm{ }}60{\rm{ }}c{m^3}$

For a cube of side $$l$$ units, its volume will be

$V = {l^3}\,{\rm{unit}}{{\rm{s}}^3}$

Example 1: The dimensions of the base of a cuboid are 5 cm × 3 cm, while its height is 2 cm. Determine the total SA (surface area), lateral SA and volume of the cuboid.

Solution: We have:

• Total SA = $$2\left( {lb + bh + lh} \right) = 62\,{\rm{c}}{{\rm{m}}^2}$$

• Lateral SA = $$2h\left( {l + b} \right) = 32\,{\rm{c}}{{\rm{m}}^2}$$

• Volume = $$lbh = 30\,{\rm{c}}{{\rm{m}}^3}$$

Example 2: The volume of a cuboid is 0.16 liters, or 0.16 L. The base dimensions of the cuboid are 8 cm × 5 cm. What is the total SA of the cuboid?

Solution: Let the height of the cuboid be $$h$$ cm. We have:

\begin{align}&V = lbh\\ &\Rightarrow \,\,\,0.16\,{\rm{L = 160 c}}{{\rm{m}}^3} = 8\,{\rm{cm}}\, \times \,5\,{\rm{cm}}\, \times \,h\,{\rm{cm}}\\& \Rightarrow \,\,\,h = 4\end{align}

The total surface area is

\begin{align}&S = 2\left( {lb + bh + lh} \right)\,{\rm{c}}{{\rm{m}}^2}\\&\,\,\,\, = 2\left( {40 + 20 + 32} \right)\,{\rm{c}}{{\rm{m}}^2}\\&\,\,\,\, = 184\,{\rm{c}}{{\rm{m}}^2}\end{align}

Example 3: A cube whose volume is 1 L has a total SA of _____ cm2.

Solution: If the length of each side of the cube is  $$x$$ cm, we have:

\begin{align}&{x^3}\,{\rm{c}}{{\rm{m}}^3} = 1\,{\rm{L}} = 1000\,{\rm{c}}{{\rm{m}}^3}\\ &\Rightarrow \,\,\,x = 10\end{align}

The total SA of the cube is $$6{l^2}\,{\rm{c}}{{\rm{m}}^2}$$ or 600 cm2.

Example 4: The bases of two cuboidal containers A and B have the dimensions 10 cm x 8 cm and 15 cm x 10 cm respectively. Water is filled in A up to a height of 15 cm. Water from A is then poured into B completely. What will be the height of the water in B?

Solution:  Let the height of the water in B be $${h_B}$$ cm. As water from A is fully transferred to B, we have:

\begin{align}&{\rm{Water\,\, volume\, in\, A = Water\,\, volume\, in \,B}}\\& \Rightarrow \,\,\,10 \times 8 \times 15 = 15 \times 10 \times {h_B}\\ &\Rightarrow \,\,\,{h_B} = 8\end{align}

The height of the water in B is 8 cm.

Example 5: A cuboidal box has base dimensions 80 cm x 40 cm and has a volume of 160 L. It needs to be painted with a special kind of paint on all the sides except its bottom. The cost of this special paint is INR 6000/m2 of area. Find the cost of this paint job.

Solution: Let the height of the box be $$h$$ cm. We have:

\begin{align}&{\rm{80}}\,{\rm{cm}}\, \times \,40\,{\rm{cm}}\, \times \,h\,{\rm{cm}} = 160\,{\rm{L}} = 160000\,{\rm{c}}{{\rm{m}}^3}\\ &\Rightarrow \,\,\,h = 50\end{align}

Now, we will find the combined area of all the faces of the cuboid excluding the bottom. If we denote this by S, we have:

\begin{align}&S = lb + 2h\left( {l + b} \right)\\&\,\,\,\, = 15200\,{\rm{c}}{{\rm{m}}^2}\\&\,\,\,\, = 1.52\,{{\rm{m}}^2}\end{align}

Thus, the total cost C of the paint job will be

\begin{align}&C = {\rm{INR}}\,6000/{{\rm{m}}^2}\,\, \times \,\,1.52{{\rm{m}}^2}\\&\,\,\,\,\, = {\rm{INR}}\,9120\,\end{align}