If ‘R’ is the radius of a hexagonal cell, then show that the area of this 
hexagonal cell is given by, Area = 2.598R2
.

To show that the area of a hexagonal cell is given by the formula:

\[
\text{Area} = 2.598 R^2
\]

where \( R \) is the radius of the hexagon, we need to derive the formula step-by-step.

### Step 1: Understanding the Geometry of the Hexagon
A regular hexagon can be divided into 6 equilateral triangles, with each triangle having side lengths equal to the radius \( R \). Thus, the total area of the hexagon is 6 times the area of one equilateral triangle.

### Step 2: Area of an Equilateral Triangle
The formula for the area \( A \) of an equilateral triangle with side length \( s \) is:

\[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2
\]

In this case, the side length \( s = R \). Therefore, the area of one equilateral triangle is:

\[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} R^2
\]

### Step 3: Area of the Hexagon
Since the hexagon is made up of 6 such equilateral triangles, the total area \( A_{\text{hexagon}} \) is:

\[
A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4} R^2
\]

\[
A_{\text{hexagon}} = \frac{6\sqrt{3}}{4} R^2
\]

Simplifying the fraction:

\[
A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} R^2
\]

### Step 4: Approximating the Constant
We now approximate the value of \( \frac{3\sqrt{3}}{2} \):

\[
\frac{3\sqrt{3}}{2} \approx 2.598
\]

Thus, the area of the hexagonal cell is approximately:

\[
A_{\text{hexagon}} \approx 2.598 R^2
\]

This completes the derivation.

### Conclusion:
The area of a regular hexagonal cell with radius \( R \) is approximately:

\[
\boxed{A = 2.598 R^2}
\]

Do you need any more details or have any questions?

---

Here are 5 related questions for further exploration:
1. How is the radius related to the side length of a regular hexagon?
2. How would the area change if the hexagon is inscribed in a circle?
3. Can you derive the perimeter of the hexagon in terms of \( R \)?
4. What is the volume of a prism with a hexagonal base and height \( h \)?
5. How does the area formula change if the hexagon is not regular?

**Tip:** When deriving areas of complex shapes, breaking them down into simpler components like triangles can make calculations much easier.

If ‘R’ is the radius of a hexagonal cell, then show that the area of this hexagonal cell is given by, Area = 2.598R².

Explore the detailed solution for deriving the area of a hexagonal cell with radius R, using geometric principles. This step-by-step guide explains how the hexagon can be divided into six equilateral triangles, leading to the formula Area = 2.598R². Suitable for high school students studying geometry.

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