Math, The Way it Should Be (Unit 2.5: Areas and Volumes)

As usual for this series, if you haven't seen them before, you should check out the first post in the series and the first post in the unit.  This unit has kind of stretched on a lot longer than I intended but it is an important one because although I'm not a big fan of geometry and trigonometry, it does set up one of my favorite topics, calculus.  In light of that, we'll have this lesson and a summary and then I want to tie up some loose ends and then we'll get to the fun stuff, though perhaps not in the way you might think.

The first thing to be covered right now is the area.  Consider the old, familiar Cartesian plane:

Now imagine drawing a shape on this plane.  It would have two dimensions.  When we talked about vectors, we talked about their length, that is, the "space" that they take up in one dimension.  When we want to talk about the "space" taken up in two dimensions, we call it the area.  Many regular shapes have special formulas to help figure out their areas and in the interest of brevity, I will allow the reader to tackle that on his or her own.  What shapes really interest me right now are rectangles and circles.  The reason is because you can make any shape with small enough rectangles or circles.  This may not be obvious but trust me, I will explain later.

Circles (more on them here) have an area proportional to the square of their radius:

\(A = \pi r^2\)

I will prove this to you later, I promise, for now, just please believe me.

Rectangles (4-sided, closed shapes with all right angles) have an area equal to the product of the length of their sides:

\(A = ab\)

Where \(a\) and \(b\) are the lengths of the two adjacent sides.  This makes a lot of sense if you go back to the explanation of multiplication earlier in the series.

Volume is the amount of space an object takes up in three dimensions.  Again, regular shapes have their own special formulas, which you can find in a number of places on the internet.  For volumes, I basically only care about the three-dimensional analog of the rectangle, which is the prism.  It is has all right angles and six faces, all of which are rectangles.  In essence, the prism is a box and the volume of the box is as follows:

\(V = abc\)

Where \(a\), \(b\), and \(c\) are the lengths of the three sides that are perpendicular to each other.  

So those are the basics of areas and volumes.  I didn't spend a whole lot of time on them because I really think that memorizing a bunch of formulas for all sorts of shapes isn't very useful because you can just reconstruct them if you know the formulas for rectangles, circles, and prisms and just one other thing (just wait for unit 4!)

Homework: No homework for this lesson because I'm a) feeling benevolent and b) want to go to bed.

-Lane