Cool shapes

The big hexagon is composed of 6 equal triangular portions, and each of that portion consists of smallest triangular (the white one) number of 7+5+3+1 = 16 pieces.

Therefore, the big hexagon has a total number of 96 pieces.

Each blue “snow flake” portion consists of 12 pieces.

So 2 blue “snow flakes” has total area of (24/96)(23.38) = 5.845 cm2.

The desired answer is $5.84$ .

Let the area of one white triangle be X

The large regular hexagon is consists of 96 small triangles each of them is equal to the white one

Then

96 X = 23.38

X = 0.24354

The 2 blue region consists of 24 of such small triangle

The area of the blue region = 24 X = 24(0.24354) = 5.845

The white triangles have the same side length as the little hexagons. It can be seen that 4 of these side lengths make up one side of the big hexagon. Since the little hexagons are 1/4 as long, they have (1/4)^2 the area of the big hexagon.

Each of the blue stars can be thought of as a little hexagon bordered by 6 white triangles, and 6 white triangles together have area equal to one little hexagon. So each blue star is equal to 2 little hexagons, or 2/16 of the total area.

With both stars, the area is 4 * 1/16 * 23.38 = 5.845 cm^2

1. The hexagon can be arranged into a rectangle to find its area.
2. Area = base * height
3. Area = (1.5 s) * (sqrt(3) s)
4. Area = 1.5 sqrt(3) s^2
5. s = sqrt(Area/(1.5*sqrt(3)))
6. s = sqrt(23.38/(1.5*sqrt(3)))
7. s = approx. 2.999
8. Each six pointed star is made of 12 equilateral triangles.
9. Each side of the small triangles is equal to 1/4 of side s.
10. ss = s/4 = 0.749
11. Small triangle area = b*h/2
12. area = ss * (ss*sqrt(3)/2) / 2
13. area = 2435416
14. area * 12 = 1 six pointed star, area * 24 = total area 15 area*12 = approx 5.84