To find the area of a regular hexagon with a radius of 4, we can use the formula for the area of a regular hexagon:

$A = \frac{3\sqrt{3}}{2} r^2$

Where:

• $A$ is the area of the hexagon,
• $r$ is the radius of the circumscribed circle (distance from the center of the hexagon to a vertex).

Given that the radius $r = 4$:

$A = \frac{3\sqrt{3}}{2} \times 4^2$

$A = \frac{3\sqrt{3}}{2} \times 16$

$A = 24\sqrt{3}$

The exact area is $24\sqrt{3}$, which is approximately:

$A \approx 24 \times 1.732 \approx 41.57$

Thus, the area of the hexagon is approximately 41.57 square units.

---

Would you like more details or have any further questions? Here are some related questions:

1. How can the formula for the area of a regular hexagon be derived?
2. What is the relationship between the side length of the hexagon and its radius?
3. How can we calculate the area of a regular hexagon if only the side length is given?
4. How does the area of a hexagon compare to other polygons like squares or triangles?
5. How can you find the area of an irregular hexagon?

Tip: To visualize a regular hexagon, think of it as being made up of 6 equilateral triangles, with each triangle having the same side length as the radius!