To check a container answer, round every value to 1 significant figure, use π ≈ 3, and compare the result with your full answer. If the two are in the same region, the answer is plausible.
This final lesson in volume and capacity works with the formulae from prisms and cylinders and cones, and with the conversions in volume units and capacity.
Why does an estimate catch so many errors?
Most wrong answers in volume are not small slips in the last digit. They are errors of size: a missing one third, a squared diameter, or a forgotten conversion. An estimate shows the size of the answer immediately.
A second check is real-world sense. A drinks can holds around 330 mL, a bucket around 10 litres and a bathtub around 200 litres. If your answer puts a bucket at 0.1 litres, something is wrong.
How to estimate
- Round each measurement to 1 significant figure.
- Use π ≈ 3, and ⅓ as it is.
- Multiply using easy numbers.
- Compare with your full answer for size and units.
- Investigate any gap larger than about 10 percent, unless the rounding was very rough.
Worked example
A cylinder has radius 4.8 cm and height 10.3 cm. Find its volume, then check it is plausible.
Step 1, full calculation: V = π × 4.8² × 10.3 = π × 23.04 × 10.3 = π × 237.312 = 745.5…, so 746 cm³.
Step 2, estimate: r ≈ 5, h ≈ 10, π ≈ 3, so V ≈ 3 × 25 × 10 = 750 cm³.
Step 3, compare: 746 and 750 agree, so the answer is plausible.
The mistake to watch for
A typical error is to multiply by 2r instead of squaring r.
Mistaken working: V = π × (4.8 × 2) × 10.3 = π × 9.6 × 10.3 = 310.6 cm³
The student doubled the radius rather than squaring it.
Compare with the estimate of 750 cm³. An answer of about 311 cm³ is less than half the estimate, which signals a problem. The correction is r² = 4.8 × 4.8 = 23.04, not 4.8 × 2.
Check yourself
1. A tank measures 1.9 m by 0.52 m by 1.1 m. A student says it holds 108.7 litres. Estimate the capacity in litres and decide whether that is plausible.
Show answer
Estimate: 2 × 0.5 × 1 = 1 m³ = 1000 litres. The exact value is 1.9 × 0.52 × 1.1 = 1.0868 m³, which is about 1087 litres. The student’s 108.7 litres is not plausible, because it is ten times too small.
2. A cone has base radius 6.1 cm and vertical height 9.8 cm. A student gets 1146 cm³. Estimate and judge.
Show answer
Estimate: ⅓ × 3 × 6 × 6 × 10 = 360 cm³. The correct value is ⅓ × π × 37.21 × 9.8 = 381.9, so 382 cm³. The 1146 is about three times too large, so the student forgot the one third.
3. A cylindrical cup has radius 4 cm and height 10 cm. Its label says 250 mL. Is that plausible?
Show answer
Estimate: 3 × 16 × 10 = 480 cm³. Exact: π × 16 × 10 = 502.6… cm³, which is about 503 mL. The label of 250 mL is not plausible for a full cup of these dimensions. It is about half.
Where this leads next
Now bring all five skills together in the volume and capacity practice set, and log any repeat mistakes in the mistake log and retest queue if you use it. The non-calculator working trainer supports the mental arithmetic behind estimating.
A teacher can check how you estimate, not only whether the final answer is right. That is the kind of feedback you get in online one-to-one Mathematics tuition.