A regular hexagon is inscribed in an equilateral triangle, as shown. If the hexagon has an area of 12, the area of this triangle is
(A) 20
(D) 18
(B) 16
(E) 24
(C) 15

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-094.jpg?height=320&width=360&top_left_y=450&top_left_x=1400)
A regular hexagon is inscribed in an equilateral triangle, as shown. If the hexagon has an area of 12 , the area of this triangle is
(A) 20
(B) 16
(C) 15
(D) 18
(E) 24

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-406.jpg?height=336&width=377&top_left_y=1418&top_left_x=1378)

## Solution

First of all, we can see that each of the smaller triangles is equilateral. Also, the side length of each of these is equal to that of the inscribed hexagon. From here, we can divide the hexagon up into six smaller triangles, identical to the white triangles at the three vertices as in the diagram above.

This means that each of the six triangles of the hexagon would have an area of 2 , meaning that the large triangle would have an area of $9 \times 2$ or 18 .

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-406.jpg?height=328&width=374&top_left_y=1842&top_left_x=1380)

ANSwER: (D)