the garden is a regular hexagon each side is 4 meters long the garden will be divided into 6 equilateral triangles what is the approximate surface area

Question

all work needs to be shown. the 6 equalteral triangles will have a different typr of flower planted in each triangle

Answer 1

[revised]
For a regular hexagon, each side, s, can be split in half (that is, s/2) to be the shortest side of a 30-60-90 right triangle:
        (s/2)*1, (s/2)*√3, (s/2)*2
 
The area of each right triangle is:  (1/2)bh = (1/2)(s/2)((s/2)√3) = (1/8)s²√3
The area of each equilateral triangle is:  2*( (1/8)s²√3 )    =  (1/4)s²√3
 
There are 6 equilateral triangles, which gives the general formula for the area of a regular hexagon:
                       6( (1/4)s²√3 )
       Area_hexagon = (3/2)s²√3
 
PLZ don't plug in any numbers until you get the general formulas.
 
    Area for each type of flower = (1/4)s²√3
                                             = (1/4)4²√3
                                             = 6.9 m²
    Area of hexagon = (3/2)s²√3
                             = (3/2)4²√3 
                             = 41.6 m²

Answer 2

Here are two solutions...

A=(1/2)(r^2)(n)(sin[360/n]) where n=6

A=(1/2)(4^2)(6)(sin60º)   (the radius of the circle circumscribed around the hexagon is 4 m because the triangles are equilateral)

A=(1/2)(16)(6)(√3/2)

A=(8)(6)(√3/2)

A=(48)(√3/2)

A=24√3

A=24*1.7320508

A=41.569219 square meters

 

The other solution...

draw the height of one of the equilateral triangles

this forms a right triangle

the hypotenuse is 4, one leg is 2 (1/2 of 4), and the other leg (the height) is unknown, call it x

4^2=2^2+x^2

16=4+x^2

12=x^2

x=√12=2√3 m is the height

A=(1/2)(b)(h)

A=(1/2)(4)(2√3)

A=4√3 square meters

there are 6 triangles

A=6*4√3

A=24√3

A=41.569219 square meters

Answer 3

The altitude of an equilateral triangle is (s√3)/2. (30-60-90 triangle)

The altitude of each triangle is (4√3)/2 or 2√3 .

The area of each triangle is (0.5)(4)(2√3), or 4√3, or approximately 6.9 m²