Suppose the volume of a cylinder with diameter $d=$11.92 ± 0.01 mm and height $h=$38.06 ± 0.02 mm. Calculating $\frac{\pi d^2h}{4}$, the volume is 4247.282773 mm^3, without rounding off. Now, I read that "The rule in multiplication and division is that the final answer should have the same number of significant figures as there are in the number with the fewest significant figures." As I understand, the volume should be reported as 4247 mm^3.
However, one colleague insists that the result should be written as 4247.28 mm^3 because otherwise we are losing information from the measurement instruments. Is this another thing to take into account when rounding values from indirect measurements?
The quote is from:
https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry/02%3A_Measurement_and_Problem_Solving/2.04%3A_Significant_Figures_in_Calculations
Best Answer
Perhaps a way forward is to estimate the error in the volume?
The fractional error in $d$ is $1/1192$ and in $d^2$ is $2/1192$, and the fractional error in $h$ is $2/3806$.
An estimate of the fractional error in the volume is $\sqrt{(2/1192)^2+(2/3806)^2}\approx 0.00176$.
Thus an estimate of the error in the volume is $0.00176 \times 4247.48 \approx 7.47$
So the volume is $\bf 4247\pm 7 \,\rm mm^3$.