How do you rewrite the slope in terms of Pi?

Question

Peace to all. I am taking a Physics course that is algebra-based and within one of our assignments, we have insert numbers into Excel and generate a number of graphs. After graphing said data (which was the easy part), we have to answer a series of questions about said graphs. One of the questions asks "how could this coefficient be written in terms of π?" I must admit I have no idea of how to do so. I am not asking for the answer but assistance on how one could obtain it. The Coefficient of the slope is "0.7854m".

I was given this hint "Rewrite the area formula =2 in terms of the diameter, =/2. Rename 2 as . Rename as y." It did not make sense to me though.

Edit: In Excel we have/had to input our own numbers labeling in the following columns "Diameter (m), Diameter^2/D^2 (m^2), Radius (m), Area (m^2)". For this particular question, I had to use data from the "Area (y-axis) Vs D^2 (x-axis)".

Area #'s are: 0.785398163, 1.767145868, 3.141592654, 4.908738521, 7.068583471, 9.621127502

D^2 #'s are 1 2.25, 4, 6.25, 9, 12.25

....after graphing I got "y=0.7854x" and r^2=1.

Answer 1

I'm assuming your graph plots the area of a circle vs the diamater squared. If this is the case, then every $x$ is interpreted as a diameter squared, so we get $$ x = d^2 $$ On the other hand, since the $y$ value is interpreted as the area of a circle, then we get $$ y =\text{Area} = \pi r^2 $$ Now, since you numerically obtained the line equation $ y=0.7854 x$, this means that the slope is just the fraction $\frac{y}{x}$, or in other words $$ \frac{y}{x} = 0.7854 \tag{1} $$ The question you ask yourself now is, can this slope $\frac{y}{x}$ be written in terms of $\pi$? The answer is yes! We can use the formulas from the beginning to get this: \begin{align} \frac{y}{x}= \frac{\pi r^2}{d^2} \overset{\color{purple}{d=2r}}{=}\frac{\pi r^2}{(\color{purple}{2r})^2}= \frac{\pi r^2}{4r^2}= \frac{\pi}{4} \tag{2} \end{align} So combining equations $(1)$ and $(2)$ you get that your slope can be written in terms of $\pi$ as $$ 0.7854 \approx \frac{\pi}{4} $$ giving the desired interpretation, which can be verified with a calculator.

Answer 2

The area of the circle is $\,A=\pi r^2\,$, or $\,A=\pi\left(\dfrac{d}{2}\right)^2=\dfrac{\pi}{4}d^2\,$ in terms of its diameter $\,d=2r\,$.

With $\,y=A\,$ and $\,x=d^2\,$ the graph of the area against the square of the diameter is $\,y=\dfrac{\pi}{4}x\,$.

• The equation represents a straight line, so it makes sense to speak of the slope of the graph (otherwise, for a non-linear graph, the slope would change between different points).

• The slope is $\,\dfrac{dy}{dx} = \dfrac{\pi}{4}\approx 0.7854\,$. Note that the slope has dimension $\,\dfrac{m^2}{m^2}=1\,$ i.e. it is dimensionless, so it is wrong to say that the slope is $\,0.7854 \color{red}{\,\text{m}}\,$.