8.3 Volume, Capacity & Surface Area
Key Takeaways
- Volume is the space inside a 3D object, measured in cubic units; a rectangular prism has V = l x w x h, which is base area times height
- Find the volume of a composite prism by splitting it into rectangular prisms and adding, or by subtracting a missing chunk from a bounding prism
- Capacity is how much a container holds: 1 cm^3 = 1 mL and 1000 cm^3 = 1 L, which links volume and capacity directly
- Surface area is the total area of all faces; for a rectangular prism, add the three different face areas and double, or trace the net
- Volume (cm^3), capacity (mL or L) and surface area (cm^2) measure different things — ICAS traps rely on you mixing up their units
Three-dimensional measurement appears from the middle papers onward and becomes a major topic by Papers G-J, where geometrical measurement and packing problems show up regularly. The good news: almost every question reduces to one formula, one conversion fact, and one careful habit of tracking units. With no personal calculator, favour problems where you multiply small numbers in a convenient order.
Volume and Cubic Units
Volume is the amount of space inside a 3D object, measured in cubic units. A 1 cm^3 cube is a cube with edges of 1 cm. Younger papers ask you to count cubes in a stack: count one layer, then multiply by the number of layers, and remember the hidden cubes you cannot see at the back.
Volume of a Rectangular Prism
V = length x width x height
Equivalently, V = (area of base) x height, which generalises to any prism: the volume is the area of the uniform cross-section times the length.
Worked Example 1
A fish tank is 50 cm long, 30 cm wide and 20 cm high. Find its volume.
V = 50 x 30 x 20. Multiply in an easy order: 50 x 20 = 1000, then 1000 x 30 = 30,000 cm^3. Estimation check: about 50 x 30 x 20 is roughly 30 thousand, so the answer size is right.
Composite Prisms
Just like composite areas, split an L-shaped solid into two rectangular prisms and add, or subtract a missing chunk from a bounding prism.
Worked Example 2
A step-shaped block is a 10 cm x 6 cm x 4 cm prism with a 4 cm x 6 cm x 2 cm notch removed. Find its volume.
Bounding prism: 10 x 6 x 4 = 240 cm^3. Notch: 4 x 6 x 2 = 48 cm^3. Volume = 240 - 48 = 192 cm^3.
Capacity: mL and L
Capacity is how much liquid (or gas) a container can hold. The metric facts to memorise:
| Capacity | Volume link |
|---|---|
| 1 millilitre (mL) | = 1 cm^3 |
| 1 litre (L) | = 1000 mL = 1000 cm^3 |
| 1 kilolitre (kL) | = 1000 L = 1 m^3 |
The bridge 1 cm^3 = 1 mL turns volume questions into capacity questions. The fish tank above holds 30,000 cm^3 = 30,000 mL = 30 L.
Worked Example 3 (filling problem)
A jug holds 2 L. How many 250 mL cups can it fill?
Convert to the same unit: 2 L = 2000 mL, and 2000 / 250 = 8 cups. Halve-and-double trick: 2000 / 250 = 4000 / 500 = 8000 / 1000 = 8.
Worked Example 4 (packing problem)
A carton is 40 cm x 30 cm x 20 cm. How many 10 cm cubes fit inside?
Do not divide volumes blindly — check the fit along each edge: 40/10 = 4, 30/10 = 3, 20/10 = 2, so 4 x 3 x 2 = 24 cubes. Dividing volumes works here (24,000 / 1000 = 24) only because the cubes fit evenly. If a box edge is 35 cm, 10 cm cubes give only 3 rows with wasted space, and the volume shortcut would overstate the answer. ICAS tests exactly this distinction.
Surface Area
Surface area is the total area of all the outside faces, measured in square units. For a rectangular prism there are three pairs of identical faces:
SA = 2 x (lw + lh + wh)
Worked Example 5
Find the surface area of a 5 cm x 3 cm x 2 cm box.
Faces: 5 x 3 = 15, 5 x 2 = 10, 3 x 2 = 6. Sum = 31, doubled = 62 cm^2.
Using Nets
Middle papers show a net (the unfolded shape) and ask for the surface area, or which net folds into a given solid. Trace each face on the net, compute its area, and add them all. For a cube of side s, SA = 6s^2: a 4 cm cube has SA = 6 x 16 = 96 cm^2. A classic trap is computing the volume (s^3 = 64) when the question asks for surface area.
Keeping the Three Ideas Apart
| Concept | What it measures | Units | Key formula/fact |
|---|---|---|---|
| Volume | space inside | cm^3, m^3 | V = l x w x h |
| Capacity | how much it holds | mL, L | 1 cm^3 = 1 mL |
| Surface area | skin of the solid | cm^2, m^2 | SA = 2(lw + lh + wh) |
Read the final line of every question twice: "how much water can it hold?" wants capacity, "how much cardboard is needed?" wants surface area, and "how much space does it take up?" wants volume. The units in the answer options will tell you which one the examiner expects — if every option ends in cm^2, you are doing a surface area question whether you realised it or not.
Mental Arithmetic Tips for 3D Problems
- Multiply in the friendliest order: pair numbers that make 10, 100 or 1000 (e.g. 25 x 4, 50 x 20).
- Keep place value tidy with zeros: 30 x 20 = 6 with two zeros = 600.
- Convert to the answer's unit at the very end if possible, so the numbers stay small along the way.
A rectangular water tank is 60 cm long, 40 cm wide and 25 cm high. How many litres of water can it hold when full?
What is the surface area of a cube with edges of 5 cm?