7.3 Volume & Surface Area
Key Takeaways
- Volume uses cubic units (cm³, in³, m³) and surface area uses square units (cm², in²) — reporting one with the other's unit is an automatic miss.
- Rectangular prism: V = length × width × height and SA = 2(lw + lh + wh); cube: V = s³ and SA = 6s²; cylinder: V = π × r² × h.
- The bridge that makes volume clinical: 1 cm³ = 1 mL exactly, so 1,000 cm³ = 1 L.
- Cubic conversions are not linear conversions — 1 m = 100 cm, but 1 m³ = 1,000,000 cm³ because each of the three dimensions is scaled.
- A cylinder needs the radius, not the diameter; halve the diameter before squaring it.
Why Volume Is on the Measurement Blueprint
Measurement is the largest content area on the NEX at 35% (14 scored items), and the NLN describes it as using units, scales, ratios, and proportional relationships and applying basic formulas to solve real-world measurement problems. The Common Core priorities the NLN cites by name include problems involving area, surface area, and volume. Section 7.2 handled the flat (two-dimensional) shapes; this section handles the solid (three-dimensional) ones.
There is also a purely practical reason to know volume: it is the hinge between geometry and every fluid calculation you have already practiced. A basin, a syringe barrel, a specimen container, and an IV bag all hold a volume, and metric volume is defined so that geometry and fluid measurement use the same numbers.
Volume: Cubic Units
Volume is the amount of space a solid occupies. Because you are filling a space in three directions — length, width, and height — the units are cubic: cm³, m³, in³, ft³. If your answer to a volume question carries a square unit, you computed an area by mistake.
| Solid | Volume Formula | Notes |
|---|---|---|
| Rectangular prism (box) | V = length × width × height | The three dimensions can be given in any order |
| Cube | V = s³ (side × side × side) | All edges are equal |
| Cylinder | V = π × r² × h | Area of the circular base times the height |
Worked example 1. A storage drawer measures 12 cm long, 8 cm wide, and 5 cm deep. Find its volume.
V = 12 × 8 × 5 = 480 cm³.
Worked example 2. A cube-shaped specimen box has edges of 4 in. Find its volume.
V = 4³ = 4 × 4 × 4 = 64 in³.
Worked example 3. A cylindrical container has a radius of 3 cm and a height of 10 cm. Find its volume (use π ≈ 3.14).
V = π × r² × h = 3.14 × 3² × 10 = 3.14 × 9 × 10 = 282.6 cm³.
Notice the order of operations: square the radius first, then multiply by π, then by the height. Squaring the product instead of the radius is the most common cylinder error.
The 1 cm³ = 1 mL Bridge
The metric system was built so that volume in cubic centimeters and volume in milliliters are the same quantity:
1 cm³ = 1 mL exactly, so 1,000 cm³ = 1,000 mL = 1 L.
This is why an older chart entry of "30 cc" means the same thing as "30 mL" — cc is simply cubic centimeters. (Use mL in written orders; "cc" is on most do-not-use abbreviation lists because it is easily misread as "u.")
Worked example 4. A rectangular wash basin measures 30 cm × 20 cm × 10 cm. How many liters does it hold when filled?
V = 30 × 20 × 10 = 6,000 cm³ = 6,000 mL = 6 L.
Worked example 5. A cylindrical specimen container has a radius of 2.5 cm and a height of 8 cm. How many mL does it hold?
V = 3.14 × 2.5² × 8 = 3.14 × 6.25 × 8 = 157 cm³ = 157 mL.
Surface Area: Square Units
Surface area is the total area of every outside face of a solid — the amount of covering material it would take to wrap it. Because you are adding up flat faces, the units are square: cm², in², m².
| Solid | Surface Area Formula | Why |
|---|---|---|
| Rectangular prism | SA = 2(lw + lh + wh) | Three pairs of matching faces |
| Cube | SA = 6s² | Six identical square faces |
Worked example 6. Find the surface area of the 12 cm × 8 cm × 5 cm drawer from Example 1.
- lw = 12 × 8 = 96
- lh = 12 × 5 = 60
- wh = 8 × 5 = 40
- Sum = 96 + 60 + 40 = 196
- SA = 2 × 196 = 392 cm²
Worked example 7. A cube has edges of 5 in. Find its surface area.
SA = 6 × 5² = 6 × 25 = 150 in².
Compare that with its volume, 5³ = 125 in³. Same solid, same edge length, two different numbers with two different units — a favourite NEX distractor pairing.
Converting Cubic Units — The Big Trap
A linear conversion factor does not carry over to volume unchanged. Because all three dimensions scale, you apply the factor three times:
| Linear | Area (factor²) | Volume (factor³) |
|---|---|---|
| 1 m = 100 cm | 1 m² = 10,000 cm² | 1 m³ = 1,000,000 cm³ |
| 1 ft = 12 in | 1 ft² = 144 in² | 1 ft³ = 1,728 in³ |
Worked example 8. A container holds 0.002 m³. How many cm³ is that?
0.002 × 1,000,000 = 2,000 cm³, which is also 2,000 mL, or 2 L.
If you had multiplied by 100 instead of 1,000,000 you would have reported 0.2 cm³ — off by a factor of 10,000. The safest habit is to convert every dimension to the target unit before multiplying, so you never have to cube a factor in your head.
Worked example 9. A box is 1.2 m × 40 cm × 25 cm. Find its volume in cm³.
Convert first: 1.2 m = 120 cm. Then V = 120 × 40 × 25 = 120,000 cm³ (= 120 L).
Common Traps
- Reporting volume in square units (or surface area in cubic units). Check the exponent on your unit before you commit to an answer.
- Using diameter as the radius in a cylinder. V = πr²h needs the radius. A diameter of 6 cm means r = 3 cm, and using 6 would quadruple your answer.
- Cubing a linear factor by accident — or failing to. 1 m³ is 1,000,000 cm³, not 100 cm³.
- Mixing units inside one formula. A box given as 1.2 m × 40 cm × 25 cm must be converted to a single unit before multiplying.
- Confusing volume with capacity units. They are the same thing in metric: 1 cm³ = 1 mL. Do not "convert" between them with any factor other than 1.
A rectangular storage bin measures 25 cm long, 20 cm wide, and 12 cm deep. What is its volume in liters?
A cube-shaped specimen box has edges of 5 in. What is its total surface area?