Question

Regular hexagon .with example

Answer 1

A regular hexagon is a polygon with six equal sides and six equal angles, where each interior angle measures \(120^\circ\). A regular hexagon is symmetric and looks the same from multiple angles.

### Properties of a Regular Hexagon:
1. **Sides and Angles**: All sides are of equal length, and all internal angles are equal (each interior angle is \(120^\circ\)).
2. **Symmetry**: A regular hexagon has 6 lines of symmetry and rotational symmetry of order 6.
3. **Area**: The area of a regular hexagon can be calculated using the formula:
\[
\text{Area} = \frac{3\sqrt{3}}{2} s^2
\]
where \(s\) is the length of a side.
4. **Circumradius**: The radius of the circumscribed circle (circumradius) is equal to the length of a side: \(R = s\).
5. **Inradius**: The radius of the inscribed circle (inradius) can be calculated as:
\[
r = \frac{s\sqrt{3}}{2}
\]

### Example:
Consider a regular hexagon with a side length of \(s = 4\) units.

1. **Calculating the Area**:
\[
\text{Area} = \frac{3\sqrt{3}}{2} \cdot 4^2 = \frac{3\sqrt{3}}{2} \cdot 16 = 24\sqrt{3} \approx 41.57 \text{ square units}
\]

2. **Calculating the Inradius**:
\[
r = \frac{4\sqrt{3}}{2} = 2\sqrt{3} \approx 3.46 \text{ units}
\]

3. **Visual Representation**:
A regular hexagon can be visually represented as follows:

```
_______
/ \
/ \
/ \
| |
\ /
\___________/
```

This illustration shows a regular hexagon where each side is of equal length, and its angles are equal, representing its symmetry and uniformity.