## Task 3 - 290623

The image shows a square composed of nine fields. The side length of each individual field is $1 \mathrm{~cm}$.

a) Determine the number of all rectangles that consist of such fields!

b) Determine the sum of the areas of all these rectangles!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0270.jpg?height=197&width=197&top_left_y=313&top_left_x=1563)
For every possible size of the mentioned type of rectangles, the following table provides the listed information:

| Rectangles | Area | Number of Rectangles | Sum of Areas |
| :---: | :---: | :---: | :---: |
| $1 \mathrm{~cm} \cdot 1 \mathrm{~cm}$ | $1 \mathrm{~cm}^{2}$ | 9 | $9 \mathrm{~cm}^{2}$ |
| $1 \mathrm{~cm} \cdot 2 \mathrm{~cm}$ | $2 \mathrm{~cm}^{2}$ | 12 | $24 \mathrm{~cm}^{2}$ |
| $1 \mathrm{~cm} \cdot 3 \mathrm{~cm}$ | $3 \mathrm{~cm}^{2}$ | 6 | $18 \mathrm{~cm}^{2}$ |
| $2 \mathrm{~cm} \cdot 2 \mathrm{~cm}$ | $4 \mathrm{~cm}^{2}$ | 4 | $16 \mathrm{~cm}^{2}$ |
| $2 \mathrm{~cm} \cdot 3 \mathrm{~cm}$ | $6 \mathrm{~cm}^{2}$ | 4 | $24 \mathrm{~cm}^{2}$ |
| $3 \mathrm{~cm} \cdot 3 \mathrm{~cm}$ | $9 \mathrm{~cm}^{2}$ | 1 | $9 \mathrm{~cm}^{2}$ |

a) The number of the mentioned rectangles is 36.

b) The sum of their areas is $100 \mathrm{~cm}^{2}$.

Solution adopted from $[5]$