Q.Each side of a cube is measured to be $7.203\ \text{m}$. What are the total surface area and the volume of the cube to appropriate significant figures?

Side $a = 7.203\ \text{m}$ has 4 significant figures, so the results must too.

$A = 6a^2 = 6 \times 51.883209 = 311.299254\ \text{m}^2 \to 311.3\ \text{m}^2$ …

Setting up

A cube has side $a = 7.203\ \text{m}$. This measurement has 4 significant figures ($7$, $2$, $0$, $3$), so any quantity calculated from it can be trusted only to 4 significant figures — no more.

Surface area

1. Square the side: $a^2 = (7.203)^2 = 51.883209\ \text{m}^2$
2. Multiply by 6 (a cube has 6 faces): $A = 6 \times 51.883209 = 311.299254\ \text{m}^2$
3. Round to 4 significant figures: the first four digits are $3,1,1,2$; the next digit is $9$, so round up — $A = 311.3\ \text{m}^2$

Volume

1. Cube the side: $V = (7.203)^3 = 373.714754\ \text{m}^3$ (extra digits kept for now) …

Applying the weakest-link rule

1. State the rule. For a product or quotient of measured quantities, the number of significant figures in the result equals the number of significant figures in the least precise input — extra digits beyond that are not physically meaningful.
2. Find the limiting precision. Here $a = 7.203\ \text{m}$ is the only measured quantity, and it appears twice (for area) or three times (for volume) in the formula. Repetition doesn't add precision — the limit stays at 4 significant figures.
3. Compute with extra digits first, round last. Don't round $a^2$ or $a^3$ before finishing:

$a^3 = 373.714754\ \text{m}^3$ …

Here are the most common mistakes students make on this Significant Figures Calculation problem, along with how to avoid each.

1. Forgetting the Rule for Multiplication

Mistake:

Students calculate the surface area or volume and then write the answer with all the digits from the calculator, e.g.,

$7.203 \times 7.203 = 51.883209$ and then they write $51.883209\ \text{m}^2$.

Why it’s wrong:

In multiplication (or division), the result should have the same number of significant figures as the measurement with the fewest significant figures. Here, $7.203$ has 4 significant figures, so the answer must also have 4 significant figures.

How to avoid:

• Always count the significant figures in each given value before calculating.
• After multiplying, round the result to match that count.
• For surface area: $6 \times (7.203)^2 = 6 \times 51.883209 = 311.299254$ → round to 4 sig figs → $311.3\ \text{m}^2$.

2. Forgetting to Apply the Rule to the Constant (like 6)

Mistake:

Students think “6” is an exact number (from geometry) and ignore it when counting significant figures — but then they still keep too many digits.

Why it’s wrong:

The constant 6 is exact (it’s not a measurement), so it does not limit significant figures. The limitation comes only from the measured side length $7.203$.

How to avoid:

• Identify which numbers are exact (counted or defined) vs. measured.
• Exact numbers have infinite significant figures — they don’t affect rounding.
• Only round based on the measured value’s significant figures.

3. Rounding Too Early in the Calculation

Mistake:

Rounding $7.203^2$ to $51.88$ (4 sig figs) before multiplying by 6, then getting $311.28$ and rounding again.

Why it’s wrong:

Rounding intermediate steps can introduce round-off error and change the final digit. The correct final answer for surface area is $311.3$, not $311.3$ from a prematurely rounded intermediate.

How to avoid:

• Keep one extra digit in intermediate steps (or use full calculator precision).
• Round only the final answer to the required significant figures.

4. Confusing Surface Area and Volume Rules

Mistake:

Using the same significant figure rule for both surface area and volume — but forgetting that volume involves three multiplications.

Why it’s wrong:

The rule is the same: the result’s significant figures = the fewest among the measured inputs. For volume:

$V = (7.203)^3 = 7.203 \times 7.203 \times 7.203$

The side has 4 sig figs, so volume must also have 4 sig figs.

How to avoid:

• For volume: $7.203^3 = 373.714...$ → round to 4 sig figs → $373.7\ \text{m}^3$. …