Volume

Key Concepts

• Volume is the amount of space an object takes up.

  • It tells us how much space is inside a solid.
  • It can also tell us how much liquid a container can hold.

• The standard unit for volume of solids is cubic units.

  • Examples:
    • cm³ = cubic centimetres
    • m³ = cubic metres
  • The word cubic means the unit is multiplied 3 times:
    • $1 \text{ cm}^3 = 1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm}$

• For liquids, volume is often measured in:

  • millilitres (mL)
  • litres (L)

• Useful conversion:

  • 1 L = 1000 mL
  • 1 cm³ = 1 mL

Unit Conversions for Volume

• 1 cm³ = 1 mL (cubic centimetres and millilitres are EQUAL)

• 1 litre (L) = 1000 mL = 1000 cm³

• 1 m³ = 1,000,000 cm³ (but this is rarely tested at PSLE)

• Common PSLE conversions:

  • A container holds 2.5 L → 2500 mL → 2500 cm³
  • A bottle holds 330 cm³ → 330 mL → 0.33 L

• Key tip: cm³ and mL are interchangeable — if a question gives volume in cm³ and asks for litres, divide by 1000.

• Worked example: “A fish tank is 40 cm long, 20 cm wide and 25 cm deep. It is filled to 80% capacity. How many litres of water does it contain?”

  • Full volume = 40 × 20 × 25 = 20 000 cm³
  • 80% = 0.8 × 20 000 = 16 000 cm³ = 16 000 mL = 16 L

Volume of Cubes and Cuboids

• A cuboid is a solid shape with:

  • 6 rectangular faces
  • a length, width, and height

• A cube is a special cuboid where:

  • all edges are equal in length
  • all faces are squares

• Formula for volume of a cuboid:

  $$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$

• Formula for volume of a cube:

  $$\text{Volume} = \text{side} \times \text{side} \times \text{side} = \text{side}^3$$

• The answer must always be written in cubic units for solids.

  • Example: $120 \text{ cm}^3$, not just 120 cm

Volume of Composite Solids

• A composite solid is a solid made by joining 2 or more simple solids together.

• In Primary 6, composite solids usually involve cuboids and cubes.

• To find the volume of a composite solid:

  1. Split the solid into smaller cubes or cuboids.
  2. Find the volume of each part.
  3. Add the volumes together.

• If the solid has a missing part or hollow part:

  1. Find the volume of the whole solid first.
  2. Find the volume of the missing part.
  3. Subtract.

• You must study the dimensions carefully.

  • Some lengths may need to be found by subtraction.
  • Example: total length = 12 cm, one part = 5 cm, so remaining length = 7 cm

Volume of Liquids

• Liquids take the shape of their container, but they still have volume.

• The volume of liquids is commonly measured using:

  • measuring cylinders
  • beakers
  • jugs
  • containers marked with mL or L

• Important ideas:

  • Capacity means the maximum amount of liquid a container can hold.
  • The amount of liquid actually inside may be less than the capacity.

• Conversion between liquid and solid volume:

  • 1 cm³ = 1 mL
  • 1000 cm³ = 1000 mL = 1 L

• Example:

  • A tank with volume $5000 \text{ cm}^3$ can hold $5000 \text{ mL}$, or 5 L

Rate of Flow Problems

• Rate of flow tells us how much liquid moves in or out in a certain time.

• It is usually measured in:

  • mL per minute
  • L per minute
  • cm³ per second

• Basic formula:

  $$\text{Rate of flow} = \frac{\text{Volume}}{\text{Time}}$$

• Rearranged formulas:

  $$\text{Volume} = \text{Rate of flow} \times \text{Time}$$
  $$\text{Time} = \frac{\text{Volume}}{\text{Rate of flow}}$$

• When solving rate of flow questions:

  • make sure the units match
  • convert when needed
    • minutes to seconds
    • mL to L
    • cm³ to mL

• In filling and emptying problems:

  • inflow means liquid enters the container

  • outflow means liquid leaves the container

  • if both happen at the same time:

    $$\text{Net rate of change} = \text{Inflow rate} - \text{Outflow rate}$$

Finding Length, Width or Height Given Volume

• If you know the volume of a cuboid and two of its dimensions, you can find the missing dimension.

• Since:

  $$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$

• To find a missing dimension:

  $$\text{Missing dimension} = \frac{\text{Volume}}{\text{Product of the other 2 dimensions}}$$

• Example:

  • Volume = 96 cm³

  • length = 8 cm

  • width = 3 cm

  • height:

    $$96 \div (8 \times 3) = 96 \div 24 = 4 \text{ cm}$$

• Always check:

  • Are the units the same?
  • Does the answer make sense?

Important Definitions

• Volume: the amount of space taken up by a solid or the amount of space inside a container.

• Cuboid: a 3D solid with 6 rectangular faces.

• Cube: a 3D solid with 6 square faces and all edges equal.

• Length: the measurement from one end of an object to the other, usually the longest side of a base.

• Width: the measurement across an object from side to side.

• Height: the vertical measurement from bottom to top.

• Composite solid: a solid made from two or more simple solids joined together.

• Capacity: the maximum volume of liquid a container can hold.

• Liquid volume: the amount of liquid in a container.

• Rate of flow: the volume of liquid moving in or out per unit time.

• Cubic centimetre (cm³): the volume of a cube with side length 1 cm.

• Millilitre (mL): a unit used to measure liquid volume.

• Litre (L): a larger unit used to measure liquid volume.

• Inflow: liquid flowing into a container.

• Outflow: liquid flowing out of a container.

• Net rate: the actual rate of change after subtracting outflow from inflow.

Worked Examples

Example 1: Volume of a Cuboid

A cuboid has:

• length = 12 cm
• width = 5 cm
• height = 4 cm

Find its volume.

Step 1: Write the formula

Step 2: Substitute the values

Step 3: Calculate

Answer

Example 2: Volume of a Composite Solid

A solid is made of 2 cuboids.

• Bottom cuboid:

  • length = 10 cm
  • width = 4 cm
  • height = 3 cm

• Top cuboid:

  • length = 6 cm
  • width = 4 cm
  • height = 2 cm

Find the total volume.

Step 1: Find the volume of the bottom cuboid

Step 2: Find the volume of the top cuboid

Step 3: Add the volumes

Answer

Example 3: Rate of Flow and Time

Water flows into a tank at 250 mL per minute.
How much water flows into the tank in 12 minutes?

Step 1: Write the formula

Step 2: Substitute the values

Step 3: Calculate

Step 4: Write the unit

Step 5: Convert if needed

Answer

Example 4: Finding a Missing Height

A cuboid has volume 180 cm³.
Its length is 9 cm and width is 4 cm.
Find its height.

Step 1: Write the formula

Step 2: Rearrange to find height

Step 3: Substitute the values

Step 4: Calculate

Answer

Common Mistakes to Avoid

• Forgetting to write cubic units for solid volume.

  • Wrong: 120 cm
  • Correct: 120 cm³

• Mixing up area and volume.

  • Area uses square units.
  • Volume uses cubic units.

• Using the wrong formula.

  • Volume of cuboid is:

    $$l \times w \times h$$

  • Not $l + w + h$

• Forgetting to convert units before calculating.

  • Example:
    • 2 L must be changed to 2000 mL if the rate is in mL/min

• Reading composite solids wrongly.

  • Missing hidden dimensions
  • Not subtracting or adding correctly

• Using the total height when only the extra height is needed, or the extra height when the total height is needed.

• Forgetting that:

  • 1 cm³ = 1 mL
  • 1000 mL = 1 L

• In flow questions, adding rates when you should subtract.

  • If water flows in and out at the same time:
    • use inflow − outflow

• Dividing wrongly when finding a missing dimension.

  • You divide the volume by the product of the other 2 dimensions.

• Leaving the answer in the wrong unit.

Exam Tips

• Underline or circle the important information:

  • dimensions
  • units
  • rate
  • time
  • words like total, remaining, capacity, filled, emptied

• For cuboid questions, write the formula first:

  • Volume = length × width × height
  • This helps you gain method marks.

• For composite solids:

  • state clearly:
    • Volume of whole solid = …
    • Volume of part A = …
    • Total volume = …
  • or
    • Volume of whole block − volume of missing block

• For liquid questions:

  • check whether the answer should be in mL or L

• For flow questions:

  • use the correct relationship:
    • Volume = rate × time
    • Time = volume ÷ rate
    • Rate = volume ÷ time

• If both filling and emptying happen:

  • write:
    • Net rate = inflow − outflow

• Always include units in every step when possible.

• After solving, ask yourself:

  • Is the answer reasonable?
  • Can a small box have a volume larger than a tank? If not, check again.

• In word problems, use mark-earning phrases such as:

  • Find the volume of each part
  • Add the volumes
  • Subtract the missing part
  • Convert to the same unit first
  • Using volume = length × width × height

Quick Summary

• Volume is the amount of space taken up by a solid.

• Volume of a cuboid:

  $$l \times w \times h$$

• Volume of a cube:

  $$\text{side}^3$$

• Solid volume is written in cubic units such as cm³ and m³.

• Composite solid volume is found by adding parts or subtracting missing parts.

• Liquid volume is measured in mL and L.

• 1 cm³ = 1 mL

• 1000 mL = 1 L

• Rate of flow:

  $$\text{Rate} = \frac{\text{Volume}}{\text{Time}}$$

• For filling and emptying together:

  $$\text{Net rate} = \text{inflow} - \text{outflow}$$

• Missing dimension of a cuboid:

  $$\text{dimension} = \frac{\text{Volume}}{\text{other 2 dimensions multiplied}}$$

• Always check units before and after calculating.

• Always write the final answer with the correct unit.