Question

Find the area of a regular hexagon with the given measurement.

2 sqrt3 apothem

A =

...?

Answer 1

41.57 unit²

Explanation:

We know,

Area of a regular hexagon =
`3* (sidelength)* (apothem)`.

The length of the apothem =
`2√(3)` units.

Since, we know, 'a regular hexagon splits into 6 identical equilateral triangles'.

As, the apothem of the regular hexagon = height of the equilateral triangle

So, height of the equilateral triangle =
`2√(3)` units.

As, in the equilateral triangle, 'One of the side length is the S, other will be
`(S)/(2)` and height is
`2√(3)` units'.

So, using Pythagoras Theorem, we have,

`hypotenuse^(2)=perpendicular^(2)+base^(2)`

i.e.
`S^(2)=((S)/(2))^(2)+(2√(3))^(2)`

i.e.
`S^(2)=(S^2)/(4)+12`

i.e.
`S^(2)-(S^2)/(4)=12`

i.e.
`(3S^2)/(4)=12`

i.e. [tex3S^2=48[/tex]

i.e. [texS^2=16[/tex]

i.e. S= 4 units

That is, the side length of the hexagon = 4 units.

Thus, the area of the hexagon is given by,

Area of a regular hexagon =
`3* (4)* (2sqrt{3})`

i.e. Area of a regular hexagon =
`12* (2sqrt{3})`

i.e. Area of a regular hexagon =
`24sqrt{3}`

i.e. Area of a regular hexagon = 41.57 unit²

Hence, the area of the regular hexagon is 41.57 unit².

Answer 2

There are 6 sides to a hexagon. 360/6 = 60. So each angle going out to each side measures 60 degrees. We can split one side in half to create a right triangle which has a 30 degree angle. The adjacent is the apothem. We want to find the opposite of this triangle because it will give us half the length of the side of the hexagon. So we can use tangent.


`\sf tan(x^o)=(Opposite)/(Adjacent)`


`\sf tan(30^o)=(x)/(2√(\sf 3))`

We know that
`\sf tan(30^o)=(1)/(√(\sf 3))`:


`\sf (1)/(√(\sf 3))=(x)/(2√(\sf 3))`

Multiply
`\sf 2√(\sf 3)` to both sides:


`\sf x=(2√(\sf 3))/(√(\sf 3))`

Simplify:


`\sf x=2`

So half the side length of the hexagon is 2. That means the entire side length is 2 * 2 = 4.

Now use the formula:


`\sf A=(1)/(2)ap`

This is used to calculate the area of any regular polygon. Where 'a' is the apothem and 'p' is the perimeter. The side length is 4, a hexagon has 6 sides, so the perimeter is 6 * 4 = 24. Plug in what we know:


`\sf A=(1)/(2)(2√(\sf 3))(24)`

Simplify:


`\boxed{\sf A=24√(\sf 3)}`