# Calculating uncertainty from significant figures of a value

A question in Giancoli's Physics for Scientists and Engineers (2. ed) has me confused. Here it is (Ch 1, Problem 3):

What is the area, and its approximate uncertainty, of a circle of radius $$2.7 \times 10^4$$ cm?

I got the correct answer of $$2.3 \times 10^9 \text{ cm}^2$$, but the uncertainty provided in the answer was $$0.2 \times 10^9 \text{ cm}^2$$. How was this uncertainty calculated? As far as I can tell, it is not possible to determine the uncertainty of a measure just from its value, because I have no idea with what device/technology the radius was measured. All we can know is that the doubtful figure is in the $$10^3$$ position, and therefore the uncertainty in the area will be $$x \times 10^9$$, but $$x$$ could be anything? Why is $$x = 0.2$$?

Most likely, the authors assume Gauss error propagation in which the error on a function $$f(x)$$ of a variable $$x$$ is calculated as $$\Delta f = \frac{\partial f}{\partial x} \Delta x~.$$ In your case, $$f(x) = \pi x^2$$ and $$\frac{\partial f}{\partial x} = 2\pi x$$, and $$\Delta x = 0.1\times 10^4$$cm (this is the worst case - realistically, we could also assume $$\Delta x = 0.05\times 10^4$$cm). Taking $$\Delta x = 0.1\times 10^4$$cm leads to $$\Delta f = 2\pi x \Delta x = 2\pi \times 2.7\times 10^4 \text{cm} \times 0.1\times 10^4\text{cm} = 0.1696 \times 10^9\text{cm}\simeq 0.2 \times 10^9\text{cm}~.$$
A rough and ready way is to say that the radius is $$2.7\pm 0.1 \times 10^4\,\rm cm$$ which is approximately a $$3.7\%$$ error.
The area is the radius squared so the error in the area is twice the error in the radius, $$2\times 3.7 = 6.4\%$$ which then translates into an error of $$2.3 \times 10^9 \times 0.064 \approx 0.2 \times 10^9\,\rm cm^2$$ in the area.