State whether the following statement is True or False and justify your answer:
The area of a regular hexagon of side $a$ is the sum of the areas of the five equilateral triangles with side $a$.
The area of a regular hexagon of side $a$ is the sum of the areas of the five equilateral triangles with side $a$.
(FALSE) The statement is False.
A regular hexagon can be divided into six equilateral triangles by joining its center to each of its vertices.
Since each of these six triangles is an equilateral triangle with side $a$, the total area of the regular hexagon is equal to the sum of the areas of these six equilateral triangles.
Therefore, the area of a regular hexagon of side $a$ is $6 \times (\text{Area of an equilateral triangle of side } a)$, not the sum of the areas of five such triangles.
A regular hexagon can be divided into six equilateral triangles by joining its center to each of its vertices.
Since each of these six triangles is an equilateral triangle with side $a$, the total area of the regular hexagon is equal to the sum of the areas of these six equilateral triangles.
Therefore, the area of a regular hexagon of side $a$ is $6 \times (\text{Area of an equilateral triangle of side } a)$, not the sum of the areas of five such triangles.
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