Question

What is the area of a hexagon with an apothem of $9$ ?

Answer 1

Hint: At first, we divide the hexagon into six equilateral triangles. As we are given the height of one triangle, we can calculate the area of it using two formulae $h={\displaystyle \frac{\sqrt{3}}{2}}a$ , and $Area={\displaystyle \frac{\sqrt{3}}{4}}{a}^2$ . Hence by multiplying the area of one triangle by six, we can get the area of the entire hexagon.

Complete step by step answer:
The apothem is the distance between the centre of a polygon and the midpoint of one of its sides. In the given problem, the polygon is a hexagon (six sides). The hexagon has several properties associated with it. One of the properties is that if we join the centre to all the vertices then we get six equilateral triangles, all of the triangles having the same area.
The apothem of a hexagon is basically the height of the equilateral triangle. We have a predefined relation between the side and height of an equilateral triangle, which is
$h={\displaystyle \frac{\sqrt{3}}{2}}a$ where “h” is the height and “a” is the side of the triangle.
Now putting the value of $h=9$ , we get the value of “a” as,
$\Rightarrow a={\displaystyle \frac{2}{\sqrt{3}}}\left(9\right)=6\sqrt{3}$
As we know the area of an equilateral triangle is,
$Area={\displaystyle \frac{\sqrt{3}}{4}}{a}^2$, hence putting the value of $a=6\sqrt{3}$ , we get the area as $27\sqrt{3}$ . Now we have a total of $6$ equilateral triangles. Hence the total area of the hexagon is $6\times 27\sqrt{3}=162\sqrt{3}$ .

Note: We should be aware of the less commonly used terms such as apothem. We should draw the diagrams carefully and carry out the various calculations attentively. At last, we should remember the number of triangles with the area of one triangle.