Question

The diagram below shows a rectangle inside a regular hexagon . the apothem of the hexagon is 19.05 units . to the nearest square unit . what is the area of the shaded region

A. 1257
B.861
C.442
D.1653

Answer 1

The correct answer is: B 861

Answer 2

B. 861.

Explanation:

In the figure attached, you can observe that the shaded area can be found by calculating the difference between the area of the regular hexagon and the area of the rectangle inside.

The area of a regular hexagon is defined as

`A_(hexagon) =(Pa)/(2)`

Where
`P` is the perimeter and
`a` is the apothem.

By given, we know that the side of the hexagon is 22 units.

Remember that the perimeter is the sum of all sides, in this case would be

`P_(hexagon)=22(6) =132`.

The perimeter of the hexagon is 132 units.

Therefore, the area is

`A_(hexagon) =(132(19.05))/(2)=(2514.6)/(2)=1257.3 u^(2)`

On the other hand, the area of the rectangle is defined as

`A_(rectangle)=bh`, where
`b` is the base and
`h` is the height. We know, by given

`b=22\\h=18`

Replacing these values, we have

`A_(rectangle)=bh=22(18)=396 u^(2)`

As we said before, the shaded are is defined by the following difference

`A_(shaded)=A_(hexagon)-A_(rectangle)`

Replacing each area, we have

`A_(shaded)=A_(hexagon)-A_(rectangle)\\A_(shaded)=1257.3u^(2) -396u^(2)=861.3 u^(2)`

Therefore, the right answer is B. 861.