A regular hexagon can be divided into six equilateral triangles. To calculate the area of the hexagon, we can use the formula for the area of one of these triangles and then multiply by six.
1. **Understanding the Radius**: In the context of a regular hexagon, the radius refers to the distance from the center of the hexagon to any of its vertices (corners). For our hexagon, this radius is given as 4 units.
2. **Finding the Area of One Triangle**: The formula to calculate the area of an equilateral triangle is:
A = (√3 / 4) * s²
where s is the length of a side. To find the side length of the triangles, we use the relationship between the radius (R) and the side length (s) of a regular hexagon:
s = R = 4
3. **Calculating the Area of the Triangle**: Now, substituting this side length back into the triangle area formula:
A = (√3 / 4) * (4)²
A = (√3 / 4) * 16
A = 4√3
4. **Calculating the Total Area of the Hexagon**: Since the area of the hexagon is the sum of the areas of the six triangles:
Total Area = 6 * (4√3) = 24√3
5. **Numerical Approximation**: To get a numerical approximation, we can use the value of √3 (approximately 1.732):
Total Area ≈ 24 * 1.732 ≈ 41.568
So, the approximate area of the regular hexagon with a radius of 4 is about 41.57 square units.