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= (3 √(3) )/(2) a^(2)`
where
A - the area of the hexagon
a - the side of the hexagon

Therefore:

`A= (3 √(3) )/(2) 12^(2)`

`A= (3 √(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)= (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)= (a*h)/(2)`

`A_(1)= (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