4. The perimeter of a square is $20 \mathrm{~cm}$. By how many square centimeters will the area of the square increase if its perimeter is increased by 12 cm?
4. Since all sides of a square are equal, then
1) $20: 4=5$ (cm) — the length of one side;
2) $5 \times 5=25\left(\mathrm{~cm}^{2}\right)$ - the area of the square;
3) $12: 4=3$ (cm) - by how much the side increased;
4) $5+3=8$ (cm) - the side of the new square;
5) $8 \times 8=64\left(\mathrm{~cm}^{2}\right)$ - the area of the new square;
6) $64-25=39\left(\mathrm{~cm}^{2}\right)$.

This problem can be solved differently.

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-017.jpg?height=469&width=529&top_left_y=1002&top_left_x=752)

To answer the question of the problem, it is necessary to find the sum of the areas of the shaded rectangles.

1) $20: 4=5$ (cm) - the length of the side of the square;
2) $5+3=8$ (cm) - the length of the upper rectangle;
3) $8 \times 3=24\left(\mathrm{~cm}^{2}\right)-$ the area of the upper rectangle;
4) $5 \times 3=15\left(\mathrm{~cm}^{2}\right)$ - the area of the lower rectangle;
5) $24+15=39\left(\mathrm{~cm}^{2}\right)$ - the area of the two rectangles.

Answer: the area increased by $39 \mathrm{~cm}^{2}$.