As I understood your question, you are concerned if we cannot sum infinitely many rectangles to get a triangle.
The thing is, to understand what infinite sum of those rectangles means, you first need to get a concept that we can't just sum and count the total area of all infinitely many rectangles but we can watch what total area equals to, as we draw many thin rectangles. For instance, if we count the total area of 100 of rectangles, we get $area = \pi r^2 - 0.004$,
then we count the total area of 1000 of rectangles, we get $area = \pi r^2 - 0.00003$, then we count the total area of 100000 of rectangles, we get $area = \pi r^2 - 0.0000000002$. So we can see that the sum approaches a value $ \pi r^2$; but as we sum all this, geometrically it means that the area of the triangle gets better and better approximated by this sum. But we could also do it with "bigger rectangles" that are slightly bigger than needed (their height is right-side value) so that the sum is getting like $ \pi r^2 + 0.00003, \pi r^2 + 0.000000001, \pi r^2 + 0.000000000000001$. So the area is guaranteed to be less than $ \pi r^2 + 0.000000000000001 $ but bigger than $ \pi r^2 - 0.0000000002$. So wouldn't it be reasonable to call the "area" of the triangle precisely equal to $ \pi r^2$?
Think about it. All the axioms by which the "area" is defined hold. Moreover, it is the only reasonable value to assign to that area: all the other numbers are bigger or less than it. That value is called the limit of the sum. Nice thing is that that limit is often easy to find, for example, we can see that $\{..., \pi r^2 + 0.00003, \pi r^2 + 0.000000001, \pi r^2 + 0.000000000000001,...\} $converges to $ \pi r^2$.
So as we have all this: $ \pi r^2$ - is only reasonable value, all axioms hold, easy to find, so we define the area of a shape as this limit. So all areas which we encounter could be thought as these limits: even the area of a 1 by 1 square: the limit of a sequence $\{..., 1, 1, 1, 1, 1, ....\} \rightarrow 1$. So this new way to define the area is an extension of our thoughts of an area, and this could be thought of as a real pure area of a shape. So the real pure area of the triangle could be only $ \pi r^2$, as all other values are clearly not exact.
What's interesting, with a similar method we define all areas under curves, length of a curly line and volumes and many many more. The limit - the thing that we can never approach by an approximation but we can easily evaluate numerically, is the thing to define all non-linearly (curvy) defined things. And it's the only reasonable technique to evaluate those, on first glance impossible to get, areas, volumes, etc.
The infinite sums are defined to be equal to the limit (in a sense, but we can set axioms so that there are no contradictions when doing "infinite summing"), and only after all this, it's perfectly reasonable to think about summing infinite number of things, however, it's quite intuitive in the first place.
P.S. For some time I also wasn't sure if number $ \pi $ as many irrational numbers are precise in a sense that we can show them in decimal only as an infinite number like $ \pi = 3.1415926...$ .But, even though we can't show it as a finite thing in decimal, it could be thought of, as a precise point on a number line, as we can define the set of real numbers in such a way. Moreover, when we show a number in a form like $3.1415926...$ we mean the limit (namely the certain point on a number line to which the sequence $\{3.14,3.141,3.1415,3.14159,...\} $of points on a number line approaches, and we know that there is one and only one number (the concrete point on a number line) to which this sequence approaches and we define it to be equal to $3.1415926...$ as it's the only reasonable value to assign to this infinite representation of a number).
All these thoughts are only what I've been thinking of because I am a calculus student just like you and for a long time I was struggling to get rid of that sense of incorrectness when talking about limits. So, in some points I can be wrong.
I hope this has helped you!