kitchen table math, the sequel: how to remember

Monday, March 28, 2011

how to remember

FedUpMom on remembering area vs circumference:
I had the following brainstorm at an embarrassingly advanced age:

For a long time, I knew there were two formulas that were somehow relevant to circles, namely 2πr and πr2, but I could never remember which one was area and which one was circumference.

I finally realized that  πr2 must be the formula for area, because area is described in square units. 
Tell your kids.


Bostonian said...

A circle of radius r can be inscribed in a square of length 2*r. The square has area 4*r^2, and the circle has area pi*r^2. Since the circle occupies most but not all of the square, this makes sense.

Even students who cannot handle a BC-level calculus course should be taught the power rule for differentiation. The circumference of a circle is given by

d/dr (pi*r^2) = 2*pi*r .

Similarly, the anti-derivative of 2*pi*r is pi*r^2.

FedUpMom said...

Bostonian, I am so close to understanding this! Can you remind me of the connections between derivative, anri-derivative, area, and circumference? Thanks.

Michael Weiss said...

Imagine a circle of a certain radius R. At this moment in time it has a circumference and an area.

Now imagine adding a very, very thin layer around the circle to make it slightly thicker. The radius increases by a tiny amount (dR). The area increases by a tiny amount, too. How much? Well, if you imagine unwrapping that area and straightening it, you will have (approximately) a rectangle, whose dimensions are just the circumference of the circle and the thickness of the layer.

In other words, the increase in area (dA) is approximately the circumference times dR. Or, to put it another way, dA/dR = circumference. This is only approximate if dR and dA are tiny but finite; as you take the limit of dR to zero, the approximation becomes exact. So for a circle, the derivative of area (with respect to radius) is circumference. The formula bears this out: The derivative of pi*r^2 (with respect to r) is 2*pi*r.

Extra credit: Write an analogous explanation for why the derivative of the volume of a sphere is its surface area,and confirm it with the usual formulas for those.

Dawn said...

Somewhat related.

In my math ignorant, pre-homeschooling days I could never remember what exactly pi was supposed to represent. 3.14? I knew it had something to do with circles...

Then I saw pi expressed as 22/7 and it's made sense forever after.

owen thomas said...

nice proto-calculus notes,
michael weiss!

here's some more elementary
remarks that may not be amiss.

draw a regular hexagon;
connect opposite corners.
we've got six equalateral triangles
arranged about a central point.
call the length of any side "1".
obviously the distance *around*
the hexagon is then "6".
now sketch the circle through
all six vertices of the hex:
it's a pretty good fit.
so the distance around
the circle is about 6 times
the radius... or *3* times
the diameter.

π -- by definition (let's say;
other definitions are possible)--
is the *exact* ratio of the
circumference to the diameter.

all by way of demonstrating that
π is a little bigger than 3.

put a *square* around a circle
to show in a similar way that
π is (much) less than 4.

one now uses some calculus
or some (much more intuitive)
reasoning to develop
A = πr^2
(the area formula)
as a theorem.

alternatively, of course, one
could define π as the area
of the unit circle and derive
C = 2πr as the theorem.

22/7 is (of course) a mere
approximation; famously
*no* "fraction" (ratio of
counting numbers) can
express π *exactly*.

Allison said...


Pi is the ratio of the circumference to the diameter. 22/7 is not pi; it's an approximation whose first few digits are 3.14.

But Pi is an irrational number. That means it can't be expressed in the form of a/b (for any integers a and b, b non zero for the fraction to be well defined.) That also means it can't be represented by any finite or repeating decimal.

Pi is also a transcendental number, which means (iirc) there is no polynomial for which Pi is a root.

It's helpful to keep in mind the picture in your head of taking a piece of string, measuring out the distance of the diameter, and then wrapping that length of string around the circumference. No matter what the size of the circle, it'll take *that same* little bit more than 3 diameters to wrap around and complete the circumference. But that "little more" is a very complicated beastie, and very beatiful.

GS said...

I tell my students all the time about the connection between πr2 and square units. It's no magic bullet. All of them understand the connection; not all of them remember it.