SOLUTION: In the regular hexagram shown at the bottom, Line AB=4 cm. The area of the hexagram, in cm^2 is: A) 48 √ 3 B) 24 √ 3 C) 18 √ 3 D) 20 √ 3 E)22 √ 3 https://ibb.co

Algebra ->  Surface-area -> SOLUTION: In the regular hexagram shown at the bottom, Line AB=4 cm. The area of the hexagram, in cm^2 is: A) 48 √ 3 B) 24 √ 3 C) 18 √ 3 D) 20 √ 3 E)22 √ 3 https://ibb.co      Log On


   

Question 1173192: In the regular hexagram shown at the bottom, Line AB=4 cm. The area of the hexagram, in cm^2 is:
A) 48 √ 3
B) 24 √ 3
C) 18 √ 3
D) 20 √ 3
E)22 √ 3
https://ibb.co/xfhPGWG
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The area of an equilateral triangle with side length x is A+=+%28sqrt%283%29%2F4%29%2Ax%5E2

For equilateral triangle ABC, we have x = 4 as the side length. The area of triangle ABC is
A+=+%28sqrt%283%29%2F4%29%2Ax%5E2

A+=+%28sqrt%283%29%2F4%29%2A4%5E2

A+=+sqrt%283%29%2A%281%2F4%29%2A16

A+=+sqrt%283%29%2A4

A+=+4%2Asqrt%283%29

The regular hexagram is composed of a regular hexagon with equilateral triangles attached to each of the six edges of the hexagon.
We can break up the regular hexagon into 6 equilateral triangles that are all congruent to one another. These 6 additional triangles are identical to the triangles that line the outside of the hexagon.

In short: we have 6+6 = 12 equilateral triangles that are identical.

Because we have 12 identical triangles, each with area A+=+4%2Asqrt%283%29, this means the total area of the regular hexagram is 12A+=+12%2A4%2Asqrt%283%29+=+48%2Asqrt%283%29

--------------------------------------------------

Answer: A) 48 √ 3