Question

Find the area of a regular hexagon with apothem 3√ 3 mm. Round to the nearest whole number.

Answer 1

A. 94 in^2

Explanation:

Answer 2

`A=54√(3)`

Explanation:

here we are going to use the formula which is

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

Where P is perimeter and A is apothem

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 a

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 = (a)/(2)`

Also OP= a

Using Pythagoras theorem ,

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

`a^2=((a)/(2))^2+(\3sqrt{3})^2`

`a^2=(a^2)/(4)+27`

Subtracting both sides by
`(a^2)/(4)`

`a^2-(a^2)/(4)=27`

`(4a^2-a^2)/(4)=27`

`(3a^2)/(4)=27`

`a^2=(4 * 27)/(3)`

`a^2=4 * 9`

`a^2=36`

taking square roots on both sides we get

`a=6`

Now we have one side as 6 mm

Hence the perimeter is

`P=6 * 6`

`P=36` mm

Apothem =
`3√(3)`

Now we put them in the main formula

Area =
`(1)/(2) * 36 * 3√(3)`

Area=
`18 * 3√(3)`

Area=
`54√(3)`