One of the ways the area of a circle is taught to secondary school students (11-16) is to cut the circle into concentric circles and then rearrange the pieces into a triangle.
\begin{align} \text{area of cirlce} &= 1/2 \times base \times height
\\&= 1/2 \times 2\pi r \times r
\\&= \pi r^2
\end{align}
Naively, I would expect this argument to generalise. For example, a rectangle with base $a$ and height $b$ could be divided into concentric rectangles and rearranged into a triangle. As before, the base of the triangle would be the perimeter of the rectangle.
\begin{align} \text{area of rectangle} &= 1/2 \times base \times height
\\&= 1/2 \times 2(a+b) \times b/2
\\&= \frac{(a+b)b}{2}
\end{align}
However, this is not the usual formula for the area of a rectangle. What is wrong with the reasoning above?
I am interested in a rigorous justification for the circle area argument and why it doesn't apply to the rectangle. But I would also like a intuitive understanding of why the argument is valid for a circle but not a rectangle (that is an argument understandable by a secondary school student).
References
Image taken from https://byjus.com/maths/area-circle/
Best Answer
The strips on a rectangle will not lie flat. If there are $n$ strips then the part of the strips come from the base will be $\frac a{2n}$ thick. But the strips from the sides will be $\frac b{2n}$ thick. Lay them flat.... you will have to average the width and length.
We can fix this. Add up the strips from the bases and you get a triangle with base $= 2a$ and height $=\frac 12b$. Now lay out the strips from the sides. You will get a triangle with base $=2b$ and a height $=\frac 12a$. So the area of the rectangle will be $\frac 12\cdot 2a\cdot \frac 12b + \frac 12\cdot 2b\cdot \frac 12 a$.
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Imagine we tried to draw an image for your rectangle idea and you chose a rectangle that was $6$inches wide by $4$ inches high and we drew $6$ of the rectangle bands. Each band makes a "picture frame". Notice that along the base the picture frame band will be $\frac 13$ of an inch thick (because there are six bands and they stack up to half the height). But along the sides the picture frame band will be $\frac 12$ of an inch thick (because there are six bands and they lean against each other to half the width).
The difference between the bands being $\frac 13$ and $\frac 12$ in places changes everything.