### Tutorial 1.6: Part 3

Dimensional analysis: Areas and Volumes

What is the area of the red squares shown in cm^{2}?

The length of one side of the red square shown is 1/10th of a dm or a cm, so the red squares have an area of 1 cm^{2}. The total area is therefore 8 cm^{2}.

What is the area of the red squares shown in m^{2}?

The length of one side of the red square shown is 1/10th of a dm or a cm, so the red squares have an area of 1 cm^{2}.

We can use dimensional analysis to find the area of 8 cm^{2} in m^{2}. There are 10^{2} cm in 1 m, so 10^{4} cm^{2} = 1 m^{2}.

\begin{align*}
& 8\, cm^2 \times \frac{1 \, m^2}{ 10^4 \, cm^2} \\
& = 8 \times 10^{-4}\, m^2
\end{align*}
What is the area of the red squares shown in cm^{2}?

The length of one side of the red square shown is 1/10th of a dm or a cm, so the red squares have an area of 1 cm^{2}. The total area is therefore 30 cm^{2}.

What is the volume of the red cubes shown in m^{3}?

The length of one side of the red cubes shown is 1/10th of a dm or a cm, so the red cubes have a volue of 1 cm^{3}. The total volume is therefore 5 cm^{3}.

What is the volume of 5 cm^{3} in m^{3} ? 100 cm = 1 m so (100 cm)^{3} = 1 m^{3} .

\begin{align*}
& 5\, cm^3 \times \frac{1 \, m^3}{ 10^6 \, cm^3} \\
& = 5 \times 10^{-6}\, m^3
\end{align*}
What is the mass of the pile of red blocks in grams? The density of the block material is 524 kg/m^{3}

The length of one side of the red cubes shown is 1/10th of a dm or a cm, so the red cubes have a volue of 1 cm^{3}. The total volume is therefore 5 cm^{3}.

We can use dimensional analysis to find the mass of 5 cm^{3}. We need the conversion between in cm^{3} and m^{3} . There are 10^{2} cm in 1 m, so 10^{6} cm^{3} = 1 m^{3}.

\begin{align*}
& 5\, cm^3 \times \frac{1 \, m^3}{ 10^6 \, cm^3} \times \frac{524 \, kg}{m^3} \times \frac{10^3 g}{kg}\\
& = 2.62 g
\end{align*}