### Tutorial 1.6: Part 3

Dimensional analysis: Areas and Volumes

What is the area of the red squares shown in cm2? 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 cm2. The total area is therefore 8 cm2. What is the area of the red squares shown in m2? 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 cm2. We can use dimensional analysis to find the area of 8 cm2 in m2. There are 102 cm in 1 m, so 104 cm2 = 1 m2.

\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 cm2? 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 cm2. The total area is therefore 30 cm2. What is the volume of the red cubes shown in m3? 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 cm3. The total volume is therefore 5 cm3. What is the volume of 5 cm3 in m3 ? 100 cm = 1 m so (100 cm)3 = 1 m3 .

\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/m3 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 cm3. The total volume is therefore 5 cm3. We can use dimensional analysis to find the mass of 5 cm3. We need the conversion between in cm3 and m3 . There are 102 cm in 1 m, so 106 cm3 = 1 m3.

\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*}