Let's solve the problems based on the uploaded image.

3) Area of the Regular Hexagon

To calculate the area of a regular hexagon:

1. A regular hexagon can be divided into 6 equilateral triangles.
2. Each side of the hexagon is given as $4 \, \text{units}$.

The formula for the area of one equilateral triangle is: $\text{Area of one triangle} = \frac{\sqrt{3}}{4} a^2$ where $a$ is the side length.

For the hexagon: $\text{Area of hexagon} = 6 \times \text{Area of one triangle} = 6 \times \frac{\sqrt{3}}{4} a^2$

Substituting $a = 4$: $\text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4} \times 4^2 = 6 \times \frac{\sqrt{3}}{4} \times 16 = 24\sqrt{3} \, \text{square units}.$

Answer: $24\sqrt{3} \, \text{square units}$.

4) Areas of the Two Circles

The area of a circle is given by: $\text{Area} = \pi r^2$

• For $C_1$, with $r = 2$: $\text{Area of } C_1 = \pi (2)^2 = 4\pi \, \text{square units}.$

• For $C_2$, with $r = 3$: $\text{Area of } C_2 = \pi (3)^2 = 9\pi \, \text{square units}.$

Correct options:

• $\text{Area of } C_1 = 4\pi$
• $\text{Area of } C_2 = 9\pi$.

5) Number of Unit Squares for the Rectangle

The area of a rectangle is calculated as: $\text{Area} = \text{Length} \times \text{Breadth}$

Given length $6 \, \text{units}$ and breadth $5 \, \text{units}$: $\text{Area} = 6 \times 5 = 30 \, \text{square units}.$

Since each unit square has an area of $1 \, \text{square unit}$, $30$ unit squares are required.

Answer: $30 \, \text{unit squares}$.

Do you have any questions or need further clarification? 😊

5 Questions for Exploration

1. How does the formula for the area of a regular hexagon generalize to any side length?
2. Why is the area of an equilateral triangle $\frac{\sqrt{3}}{4} a^2$?
3. What would be the area of a hexagon with a side length of $6 \, \text{units}$?
4. Can the area of a circle be approximated using regular polygons?
5. What is the relation between the radius and circumference of a circle?

Tip: Remember that breaking complex shapes into smaller basic units like triangles or squares often simplifies calculations.