Question

Find the area of a regular hexagon with the given measurement. 48-inch perimeter

Answer 1

`96√(3)` inches

Explanation:

Here we are going to use the formula which is

Area=
`(1)/(2) * P * A`

Where P is perimeter and A is apothem

Here we are given that the Perimeter is 48 inches: Where perimeter is givenas

P=6s

Where s is the side of the hexagon

6s=48

s=8 inches

Please refer to the image attached with this :

In a Hexagon , there are six equilateral triangle being formed by the three diagonals which meet at point O.

Consider one of them , 0PQ with side "s"

As Apothem is the Altitude from point of intersection of diagonals to one of the side. Hence it divides the side in two equal parts . hence

`PR = (s)/(2)`

Also OP= s

Using Pythagoras theorem ,

`OP^2=PR^2+OR^2`

`8^2=((8)/(2))^2+a^2`

`8^2=4^2+a^2`

`64-16=a^2`

`a^2=48`

`a=4√(3)`

Hence We have Apothem
`a=4√(3)`

also we have the perimeter as 48

Now we put them in the main formula

Area =
`(1)/(2) * 48 * 4√(3)`

Area=
`24 * 4√(3)`

Area=
`96√(3)`