Dad & kid Exploring Areas of Rectangles and Squares.
Recently, I did a small exploration of the areas of rectangles and squares with my 7-year-old kid who loves learning about math. Here we consider rectangles with two positive integer side lengths, \(a\) and \(b\). The area of a rectangle is \(A = a \cdot b\), and its circumference is \(C = 2a + 2b\). Together, we computed \(A\) for various \(a\) and \(b\) with increasing square side \(s\).
| \(s\) | \(a\) | \(b\) | \(A\) | Comment |
|---|---|---|---|---|
| 2 | 1 | 3 | 3 | One less than square |
| 2 | 2 | 2 | 4 | Square! |
| 2 | 3 | 1 | 4 | Same as first row! |
| 3 | 1 | 5 | 5 | Four less than square |
| 3 | 2 | 4 | 8 | One less than square |
| 3 | 3 | 3 | 9 | Square! |
| 4 | 1 | 7 | 7 | Nine less than square |
| 4 | 2 | 6 | 12 | Four less than square |
| 4 | 3 | 5 | 15 | One less than square! |
| 4 | 4 | 4 | 16 | Square! |
I should note, the basic observations here were done by someone who only knows basic arithmetic operations and is curious. I added some formula to explain the ideas better to myself.
Observations
My kid made some exciting observations:
- Given a fixed circumference, the area \(A\) is largest when the rectangle is a square, that is, when \(a = b = s\).
- When the sides are one unit shorter and one unit longer than \(s\), the area is always one unit smaller than \(s^2\).
First observation
Let’s investigate the first observation. Suppose the sides of the rectangle are \(a = s - n\) and \(b = s + n\), where \(n\) is an offset such that \(|n| < s\). With this setup, we have: \[ \begin{aligned} C = 2(s - n) + 2(s + n) = 4s \\ A = (s + n)(s - n) = s^2 - n^2 \end{aligned} \] Since \(s^2\) and \(n^2\) are positive, the area \(A\) is largest when \(n = 0\), which proves the first observation.
Second observation
Now, let’s examine the second observation. The difference in areas between a square and a rectangle with sides \(s - n\) and \(s + n\) is: \[ \Delta A = s^2 - (s + n)(s - n) = s^2 - (s^2 - n^2) = n^2. \] So, indeed, for \(n = 1\), the difference in area is \(1\).
Final observation
Finally, let’s consider also negative \(n\), and examine \(A\) for \(s=4\) and \(s=5\):
| n | a | b | A | s=4 chart |
|---|---|---|---|---|
| -3 | 1 | 7 | 7 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| -2 | 2 | 6 | 12 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| -1 | 3 | 5 | 15 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\) |
| 0 | 4 | 4 | 16 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\) \(\blacksquare\) |
| 1 | 5 | 3 | 15 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\) |
| 2 | 6 | 2 | 12 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| 3 | 7 | 1 | 7 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| n | a | b | A | s=5 chart |
|---|---|---|---|---|
| -4 | 1 | 9 | 9 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| -3 | 2 | 8 | 16 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| -2 | 3 | 7 | 21 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| -1 | 4 | 6 | 24 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\) |
| 0 | 5 | 5 | 25 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\) \(\blacksquare\) |
| 1 | 6 | 4 | 24 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\) |
| 2 | 7 | 3 | 21 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| 3 | 8 | 2 | 16 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
| 4 | 9 | 1 | 9 | \(\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\) |
We chose to call this a skyscraper plot (we drew ours vertically). We observed that this skyscraper is actually a bunch of squares stacked. First, a 1x1 square then a 3x3, then 5x5, and so on. With this we can derive a fun formula for the area of any skyscraper:
\[ A(s) = \sum_{n=1}^{s} (2n-1)^2 = \frac{1}{3}s(4s^2 - 1)\,. \]
(Simplified formula from OEIS).
This simple problem was a lot more fun than I expected, and I found the final formula surprisingly beautiful. The cover image on top of this post, is a colourful rendition of above chart.
Appendix
On Mastodon, Matthias Giger made a Snap version of the cover image: Square Tower.
