A star is made by overlapping two identical equilateral triangles, as shown. The entire star has an area of 36 . What is the area of the shaded region?
(A) 24
(B) 18
(C) 27
(D) 33
(E) 30

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-066.jpg?height=290&width=252&top_left_y=1682&top_left_x=1294)
Since the two large triangles are equilateral, then each of their three angles equals $60^{\circ}$.

Therefore, each of 6 small triangles in the star has an angle of $60^{\circ}$ between the two equal sides. But each of these 6 small triangles is isosceles so each of the remaining two angles must equal $\frac{1}{2}\left(180^{\circ}-60^{\circ}\right)$ or $60^{\circ}$.

Therefore, each of the small triangles is equilateral.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-290.jpg?height=287&width=252&top_left_y=2082&top_left_x=991)

This shows us that the inner hexagon has all sides equal, and also that each angle is $180^{\circ}-60^{\circ}$ or $120^{\circ}$, so the hexagon is regular.

Next, we draw the three diagonals of the hexagon that pass through its centre (this is possible because of the symmetry of the hexagon).

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-291.jpg?height=296&width=266&top_left_y=150&top_left_x=992)

Also, because of symmetry, each of the angles of the hexagon is split in half, to get $120^{\circ} \div 2=60^{\circ}$. Therefore, each of the 6 new small triangles has two $60^{\circ}$ angles, and so must have its third angle equal to $60^{\circ}$ as well. Thus, each of the 6 new small triangles is equilateral.

So all 12 small triangles are equilateral. Since each has one side length marked by a single slash, then these 12 small triangles are all identical.

Since the total area of the star is 36 , then the area of each small triangle is $36 \div 12=3$.

Since the shaded area is made up of 9 of these small triangles, its area is $9 \times 3=27$.

ANsWER: $(\mathrm{C})$