Capacity and Mass

A 1-litre water bottle has a mass of 1 kg when full. A swimming pool holds 250,000 litres of water. Make a prediction: what is the mass of all that water? Write your reasoning below.

All three measurement systems use factors of 1000 between steps, and the rows align:

Because each linear dimension scales by a factor, the cubic units scale by the cube of that factor:

• $1\ \text{m} = 100\ \text{cm}$, so $1\ \text{m}^3 = 100^3\ \text{cm}^3 = 1{,}000{,}000\ \text{cm}^3$
• $1\ \text{cm} = 10\ \text{mm}$, so $1\ \text{cm}^3 = 10^3\ \text{mm}^3 = 1000\ \text{mm}^3$

Converting down (to smaller units): multiply. Converting up (to larger units): divide.

Example: 2.5 m³ to cm³: $2.5 \times 10^6 = 2{,}500{,}000\ \text{cm}^3$

Example: 45,000 cm³ to m³: $45{,}000 \div 10^6 = 0.045\ \text{m}^3$

Key trap: Students write 1 m³ = 100 cm³ because 1 m = 100 cm. Wrong! The exponent 3 applies to the factor: $(100)^3 = 1{,}000{,}000$.

Capacity conversions all use a factor of 1000:

• $1\ \text{kL} = 1000\ \text{L}$
• $1\ \text{L} = 1000\ \text{mL}$

Bridging to volume: $1\ \text{mL} = 1\ \text{cm}^3$, so $1\ \text{L} = 1000\ \text{cm}^3$ and $1\ \text{m}^3 = 1000\ \text{L} = 1\ \text{kL}$.

Example: Tank $20 \times 15 \times 10$ cm. $V = 3000\ \text{cm}^3 = 3000\ \text{mL} = 3\ \text{L}$.

Shortcut: cm³ to L: divide by 1000. cm³ to mL: no conversion needed (they are equal).

Mass units: $1\ \text{t} = 1000\ \text{kg} = 1{,}000{,}000\ \text{g}$.

For water only: $1\ \text{mL} = 1\ \text{g}$, $1\ \text{L} = 1\ \text{kg}$, $1\ \text{m}^3 = 1\ \text{t}$.

Example: 8 L of water: $8 \times 1\ \text{kg} = 8\ \text{kg}$.

Example: 2500 mL of water: $2500\ \text{g} = 2.5\ \text{kg}$.

Important: Only water has density 1 g/cm³. Other liquids have different densities, so the mass rule does not apply to them.

For water in a container, chain the conversions step by step:

1. Volume (cm³) → Capacity (mL): 1:1
2. Capacity (mL) → Capacity (L): ÷ 1000
3. Capacity (L) → Mass (kg): × 1
4. Mass (kg) → Mass (t): ÷ 1000

Example: Box 50 × 40 × 30 cm full of water.

$V = 60{,}000\ \text{cm}^3 = 60{,}000\ \text{mL} = 60\ \text{L} = 60\ \text{kg}\ \text{water}$

Quick chain: cm³ = mL ÷ 1000 = L = kg (water) ÷ 1000 = t

No calculator needed for most. Write units in every answer!

1. 1 5000 mL = ? L

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   5 L

2. 2 3 L = ? mL

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   3000 mL

3. 3 4500 cm³ = ? mL = ? L

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   4500 mL = 4.5 L

4. 4 0.5 m³ = ? L

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   0.5 × 1000 = 500 L

5. 5 Mass of 8 L of water?

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   8 kg

6. 6 2500 mL of water → ? g and ? kg

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   2500 g = 2.5 kg

7. 7 Box 20×10×15 cm: capacity in L?

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   V = 3000 cm³ = 3000 mL = 3 L

8. 8 3.5 m³ = ? cm³

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   3.5 × 10&sup6; = 3,500,000 cm³

9. 9 6 m³ of water → mass in t?

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   6 t

10. 10 250,000 L = ? m³ = ? kL

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    250,000 ÷ 1000 = 250 m³ = 250 kL