Question

find the area of a regular hexagon with an apothem 10.4 yards long and a side 12 yards long. round your answer to the nearest tenth. show all your work

Answer 1

The answer is 374.4 yards.

There are 2 ways to calculate the area of a a regular hexagon.
It is given:
apothem: b = 10.4 yards
side: a = 12 yards

1. The direct way is to use the formula for the regular hexagon without using an apothem:
$A= \frac{3 \sqrt{3} }{2} a^{2}$
where
A - the area of the hexagon
a - the side of the hexagon

Therefore:
$A= \frac{3 \sqrt{3} }{2} 12^{2}$
$A= \frac{3 \sqrt{3} }{2} 144$
$A= 374.4$

2. The indirect way is to sketch the hexagon with 3 diagonals and create 6 triangles. Then, it is necessary to calculate the area of one triangle and multiply it by 6. In this triangle, apothem is actually the height.
The area of the hexagon is:
A = 6 · A₁
where:
A - the area of the hexagon,
A₁ - the area of the triangle

$A_{1}= \frac{a*h}{2}$

where
h - height of the triangle.


Since apothem (b) of the hexagon is the height (h) of the triangle, then:
$A_{1}= \frac{a*h}{2}$
$A_{1}= \frac{12*10.4}{2}$
$A_{1}= 62.4$

Thus, the area of one triangle is 62.4 yards.

To calculate the area of the hexagon, we will multiply it by 6:
A = 6 · A₁
A = 6 · 62.4
A = 374.4 yards

In both cases, the result is 374.4 yards.The answer is