# 6. Clone 1

The figure shows a hexagon composed of identical equilateral triangles, each with an area of 10. Find the area of the shaded part.

![](https://cdn.mathpix.com/cropped/2024_05_06_47331c04b304090cfbafg-06.jpg?height=385&width=440&top_left_y=567&top_left_x=862)

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# Answer: 110

## Solution

We will call the equilateral triangles that make up the original hexagon unit triangles. Consider each of the highlighted triangles inside the hexagon separately. The smallest one coincides with the unit triangle and has an area of 10. The medium triangle is inscribed in a hexagon made up of six unit equilateral triangles. The diagram shows that this triangle is equal to half the area of the hexagon, as it can be divided into parts equal to the unshaded parts of this hexagon. Thus, its area is equal to three areas of the unit triangles.

![](https://cdn.mathpix.com/cropped/2024_05_06_47331c04b304090cfbafg-06.jpg?height=411&width=466&top_left_y=1599&top_left_x=858)

The largest triangle is inscribed in a figure made up of 13 unit triangles. Notice that it is also composed of an integer number of unit triangles. If we do not consider the central triangle, the rest can be divided into fragments equal to the unshaded parts. Then its area is equal to \((13-1) / 2 + 1 = 7\) areas of the unit triangles.

![](https://cdn.mathpix.com/cropped/2024_05_06_47331c04b304090cfbafg-06.jpg?height=380&width=434&top_left_y=2397&top_left_x=868)

Thus, the total area of the shaded part is \(10 \cdot (1 + 3 + 7) = 110\).

## Clone 2

In the diagram, a hexagon is composed of identical equilateral triangles, each with an area of 2. Find the area of the shaded part.

![](https://cdn.mathpix.com/cropped/2024_05_06_47331c04b304090cfbafg-07.jpg?height=385&width=440&top_left_y=573&top_left_x=862)

Answer: 22

## Clone 3

In the diagram, a hexagon is composed of identical equilateral triangles, each with an area of 3. Find the area of the shaded part.

![](https://cdn.mathpix.com/cropped/2024_05_06_47331c04b304090cfbafg-07.jpg?height=386&width=440&top_left_y=1366&top_left_x=862)

Answer: 33

## Clone 4

In the diagram, a hexagon is composed of identical equilateral triangles, each with an area of 5. Find the area of the shaded part.

![](https://cdn.mathpix.com/cropped/2024_05_06_47331c04b304090cfbafg-07.jpg?height=386&width=440&top_left_y=2154&top_left_x=862)

Answer: 55

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