To calculate the area of a regular hexagon (a hexagon with all sides equal and all internal angles equal), there are two common methods depending on the information you have:

Method 1: Using the side length ((s))

A regular hexagon can be divided into 6 congruent equilateral triangles (all sides = (s)).

1. Area of one equilateral triangle:
   [ \text{Area of triangle} = \frac{\sqrt{3}}{4}s^2 ]
2. Total area of the hexagon:
   Multiply by 6 (since there are 6 triangles):
   [ \text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2 ]

Method 2: Using the apothem ((a))

The apothem is the perpendicular distance from the center of the hexagon to any side.

1. Relate apothem to side length:
   For a regular hexagon, the apothem (a = \frac{\sqrt{3}}{2}s) → (s = \frac{2a}{\sqrt{3}}).
2. Perimeter of the hexagon:
   [ \text{Perimeter} = 6s = 6 \times \frac{2a}{\sqrt{3}} = 4\sqrt{3}a ]
3. Area using perimeter and apothem (general formula for regular polygons):
   [ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2} \times 4\sqrt{3}a \times a = 2\sqrt{3}a^2 ]

Example Calculation

If the side length (s = 2):
[ \text{Area} = \frac{3\sqrt{3}}{2}(2)^2 = \frac{3\sqrt{3}}{2} \times 4 = 6\sqrt{3} \approx 10.39 ]

If the apothem (a = \sqrt{3}):
[ \text{Area} = 2\sqrt{3}(\sqrt{3})^2 = 2\sqrt{3} \times 3 = 6\sqrt{3} \approx 10.39 ]

Both methods give the same result! Choose the one that fits the data you have. ✔️
Let me know if you need help with a specific problem.

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