Puffy Triangles

A few months ago, I was working on a lesson on sectors of circles. A sector of a circle is a piece of it cut off by slicing along two radiuses from the center.

There is a formula for the area of a sector that boils down to $$\text{(area of sector)}=\text{(area of the whole circle)}\times\text{(fraction of the circle that the sector takes up)},$$ or

$$\text{(area of sector)}=[\pi\times\text{(radius of circle)}^2]\times\frac{\text{(central angle)}}{360^{\circ}}.$$  If we rearrange the factors in that formula, we get another formulation: $$\text{(area of sector)}=\dfrac{1}{2}\times\text{(length along circumference)}\times\text{(radius of circle)}.$$ This version of the formula is reminiscent of the area formula for a triangle: $$\text{(area of triangle)}=\dfrac{1}{2}\times\text{(base)}\times\text{(height)},$$ so long as we interpret the ‘base’ to be the curvy side of the sector, which causes the ‘height’ to be equal to the radius of the circle:

This interpretation somehow feels right, especially since the ‘height’ of this sector ends up being perpendicular to the ‘base’ (as all radiuses of a circle are perpendicular to the edge), exactly as happens in triangles.

Is there any more to this idea, I wondered. Sectors and triangles both have three sides, and their area formulas seem to match. Is it possible that sectors are just triangles with one side bulging out? Are sectors just puffy triangles?

These are the types of questions that fuel mathematics. Observing that more complicated figures obey the same rules as simpler figures that we are used to enables us to study them with familiar tools. Advanced mathematics is filled with just this sort of thing. So how can we try to answer this question?

If we are to say that sectors are just puffy triangles, then it must be the case that they obey all of the rules and patterns that we already know about triangles. So we can begin our inquiry by listing some other properties of triangles and seeing whether they also hold for sectors.

Suppose we wanted to compute the area of this triangle:

We know the $A=\dfrac{1}{2}\times b\times h$ formula, but we have some options about how to measure the base and height. We could use the bottom side as the base…

…or the left side…

…or (as most people never learn) not along any side at all, so long as the base and height measure the extent of the triangle in their directions.

Can we do the same with sectors? Can we let any side be the base?

Unfortunately, this is where the similarities seem to end. If we take the ‘base’ of the sector to be a radius, then the $\dfrac{1}{2}\times b\times h$ formula computes the area of an actual triangle, whose three sides are the two radiuses and the blue segment below. The puffy part of the sector never gets counted, and so the formula would give the incorrect area.

Okay, so it looks like we can’t just think of sectors as puffy triangles. But that doesn’t mean that our inquiry was fanciful or superfluous. The similarity between these figures is too appealing to dismiss so soon as coincidence.

It turns out that there is a very good reason that those area formulas seem to match. We can take any sector and then slice it up into a bunch of thin sectors. Thin sectors look a lot like triangles, since their curvy sides are not long enough to bend much.

 

 

The area of the sector is the sum of the thin triangles, whose area we compute, of course, with the triangle area formula. What we get is that

$$\text{(area of sector)} \approx \text{(area of thin triangle 1)}+\text{(area of thin triangle 2)}+\cdots+\text{(area of thin triangle }n)$$

$$\phantom{\text{(area of sector)}} \approx \frac{1}{2}\times\text{(radius)}\times\text{(base 1)}+\frac{1}{2}\times\text{(radius)}\times\text{(base 2)}+\cdots+\frac{1}{2}\times\text{(radius)}\times\text{(base }n).$$

By factoring out $\frac{1}{2}\times\text{(radius)}$, we see that $$\text{(area of sector)} \approx \frac{1}{2}\times\text{(radius)}\times[\text{(base 1)}+\text{(base 2)}+\cdots+\text{(base }n)]$$ $$\phantom{\text{(area of sector)}} \approx \frac{1}{2}\times\text{(radius)}\times[\text{length along circumference}],$$ which is the familiar area sector formula. Although this computation was only an approximation (hence the $\approx$ instead of $=$), if we slice the sector into thin enough triangles, we can get the sides to be equal to each other through the concept of a limit.

So, the fact that these formulas resemble each other is not entirely a coincidence. While sectors are not just puffy triangles, the fact that they can be sliced into thin triangles means that they will behave like triangles in certain respects. If you’d like to see more about how, pick up any calculus book.