A Simplified Solution

Continuing my attempt to catch up on blogs I've meant to write in the past, I've one that led to quite a family discussion between myself and my parents a few weeks back. The topic was raised of students having difficulties learning mathematics, because the subject matter terrified the teachers.  Apparently early math concepts frighten many math teachers just as much as they do students. Lets look, for example, at a word problem where the objective is to calculate the area of a pizza.

First lets break this down to see its steps, and figure out why this is frightening. First, we assume the pizza is a circle. Then, we must know, or look up that the area of a circle is described as A = πr^2. (Unfortunately, I can not type superscripts here, or don't know how, so we will have to tolerate the "^" notation.) From there, one must know that π is no more than a constant number and they must calculate the radius of the circle. Then, plug and chug. However, that is the simple point of view. We can also look at the equation as such:


Obviously, A is the area, our goal, what we are trying to identify. But as for π? Pi is a symbol. It is not recognizable as a number or a letter. In fact, we are now introducing an entire new lever of language skills required to complete this problem. And lets be honest... Greek is not the most heavily taught language in schools. Then we must deal with r. When was the last time anyone has ever ordered a 6 inch radius pizza? (The answer is likely frequently... just in different words). But NO! Everyone orders 12 inch diameter pizzas (assuming you want that size). So what's with this r nonsense, when it is clear that radius is not a commonly used measurement to describe a pizza. Lastly, we started this all out by knowing that our pizza, like most pizzas, are rounds... like... a circle. And yet, now we have to square the radius? What does that even mean. How do you square a circle? And how did a square come into the picture at all?!?! 


At this point in our discussion (we had not talked about all this in so much detail... but we'll go with it), I claimed that those who feared solving for the area of a circle had no perspective. I will now take some time to "edit" and "simplify" the area of the circle. Then, perhaps those who fear this clear formula will have a bit more perspective.

Let us begin by dealing with the objective: area. Area of what? The sheer vagueness of the letter A could lead many people astray. It could be our pizza, thought what if our pizza is square? Then we're clueless. So, we will clarify by introducing a subscript that labels the area. (Unfortunately, like superscripts I don't know how to do subscripts here so we will use underscores (_).) Now, we will be working towards solving for A_c. Much better!

Next, let us deal with the radius, r. Since we've reviewed that r is not the best unit to describe a circle, we will look at d the diameter. So, in place of r we will use (d/2). Therefore, when the diameter is known, the formula becomes plug and chug once more, with no thinking involved. This is a wonderful improvement to r where we had to do division simply to come up with the number required by the formula.

Now to simplify the confusion of shapes. We don't like dealing with squares when we're talking about circles, because lets face it.... that's just confusing. So, we know that to "square a number" means to multiply it my itself. So, rather than squaring r, or as we have now simplified it, (d/2), we will rewrite r^2, as (d/2)^2, which we can further simplify to: (d/2)(d/2). Then, to make it a single fraction so there is less to look at, yet avoiding any "squared" notation, we henceforth be using (d*d)/4 (note the * represents multiplication, for lack of a more clear symbol).

Lastly, we must remove π from the equation. Euler, one of the greatest mathematicians of all time, came up with an identity, which to this day, I do not fully understand. But, I do know what it is. Euler's Identity is e^(iπ) = -1. So, if we are to proceed with our equation to a point of having no Greek letters in it, we can write an equivalent statement in terms of π. Then we find that, π = ln(-1)/i. Now, when we substitute this into our area of the circle, we have gotten rid of all characters belonging to foreign languages and don't have to deal directly with any irrational numbers.


So, at long last we have a new and "simplified" area of a circle:


A_c = (ln(-1)*d*d)/(4i)

So, to anyone who previously thought the area of a circle was terrifying, I have done my best to simplify it, and remove from the equation the elements that I thought would be the basis for confusion. I hope this helps and that you look forward to determining the area of the next pizza you buy!