Many can recite the rules to finding the area and perimeter of triangles, rectangles, and circles, but not all actually understand why these formulas work. Let’s start with the rectangle because it will actually inform the area formulas for triangles and, oddly enough, circles.
Let’s say we have the rectangle below:
We claim this rectangle has an area of 6 square units, meaning it is made of six squares that have dimensions 1 unit by 1 unit. Turns out that the length, 3 units, tells us the number of squares per row and the width, 2 units, tells us the number of rows, therefore the area of a rectangle is simply:
where A is the area, l is the length, and w is width. And perimeter (P) is just the distance around the entire rectangle which is length plus width plus length plus width, or:
Unfortunately, things get a bit more complex with triangles because there are many variations, as seen below:
Thankfully, mathematicians seek to simplify every situation as much as possible, therefore we can create a single rule for the area and a single rule for the perimeter of all triangles. First, perimeter simply means adding up the length of each side of a triangle just like the rectangle. But let’s take a deeper look at the area, taking a look at each case separately, starting with the right triangle:
Notice how the area of the right triangle is just half the area of the rectangle, therefore the formula must be:
I’m using b and h in the second formula because mathematicians typically refer to our measurements as the base and height of the triangle rather than the length and width, respectively. So in the case above, the area must be 3 square units (please check the math yourself!).
Now, I know many of you are thinking this cannot possibly work for the other cases, but I assure you it does. Let’s take a look at the following:Now, let’s add some familiar shapes:
So the area of the left triangle must be 1 square unit and the right triangle must be 3 square units, giving the whole triangle an area of 4 square units. Now, notice that our formula would have given us the same answer:
Mind blown, right?!? But, really this is just an algebraic trick using the following image as an example:
Really, the triangle area should be equivalent to the area of triangle 1 (T1) plus the area of triangle 2 (T2), therefore:
And there you have it! As long as you know the height of the triangle and the length of its base, then you can find it’s area.
We will wrap up this page with the most difficult situation. Beginning with perimeter, or what we call circumference when dealing with circles, mathematicians found that no matter what, when the circumference (C) of a circle was divided by its radius (r), the result of always 2π (remember that the radius is the distance from the center to the edge of a circle). In other words:
Now, let’s look at the circle below with radius 1:
Based on our work above, the circumference of this circle must be 1π square units. Not so bad right? But area gets much more complicated. Let’s begin by breaking up this circle into six wedges:And now, let’s rearrange them:
This probably just looks more confusing, but notice how half the circle arcs are on top and half are on the bottom. This will be important later. First, let’s break the circle into 24 wedges instead, then rearrange them:And let’s just keep doing this:
And, if I imagined I could keep doing this forever, eventually the wedges would create a perfect rectangle. Since half of the arcs are on top, the length of this rectangle must be half the circumference, or 1π units. And the width must be the same as the radius, or 1 unit. And finding the area of the rectangle we get 1π square units:With this in mind, imagine we don’t know the radius (r), but the same rules would apply, thus:And there we have it! This formula gives us the area for any circle. Now take a look at some of the possible research questions that arise from area and perimeter.:
- Irregular Features: Not all shapes are circles or polygons, therefore students will be challenged to find methods for approximating area. Google Earth has an interesting tool for students to play with.
- Gauss Circles: Moving beyond applications, here is a pure mathematics investigation into another open problem.