You don't need all those angles you show in the 2nd link.  You just need to
learn about two kinds of special right triangles, the 45-45-90 right triangle
and the 30-60-90 right triangle.  That's what this kite is made up of, 2
of each.  There are two 30-60-90 right triangles on the left, and two 45-45-90
right triangles on the right:



You should learn the following information about these
two special right triangles.  They occur very often.

The sides of the standard 45-45-90 right triangle:
The two legs are 1 unit long each.
The hypotenuse is  units long.
The triangle is both a right triangle and also an isosceles triangle.

The sides of the standard 30-60-90 right triangle:
The shorter leg is 1 unit long. 
The longer leg is  units long.
The hypotenuse is 2 units long.

Again, I emphasize that you need to learn the above information about 
those two standard right triangles. They occur often in these
geometry problems.

Let the longer leg of your 30-60-90 right triangle be x.

You are given that the shorter leg of the 30-60-90 triangle
on the upper left is given as 8 cm.

So make the proportion:





cross-multiply:





So the longer leg of each of the 30-60-90 right triangles on the left
have base  and given height 8.  So the area of each of 
those two 30-60-90 right triangles is:





Now do the same for each of the two 45-45-90 right triangles on the right.
There's less work to do here, because you know that in a 45-45-90 triangle
both legs are the same length. So we know that the base and height are
the same length.  So they are both 8 cm.  





Since the kite is made up of two 30-60-90 right triangles plus
two 45-45-90 right triangles the total area is:





and if you like, you can factor out 64, and get



That's about 

Edwin

Two triangles with the angles of 45 degrees form a square with the side of 8 cm,
if to attach them hypotenuse to hypotenuse.


So, the total area of these two triangles is   = 64 cm^2.


The great triangle on the left is the equilateral triangle with the side length a = 16 cm,
since all three its angles are 60 degrees each.


The area of such a triangle is  

     =  =  =  cm^2.


Therefore, the total area of the given kite is

    64 +  =  = 174.85 square centimeters approximately.    ANSWER

Here is a link to a good lesson on the special right triangles:

https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/4.3/primary/lesson/special-right-triangles-pcalc/

Edwin