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مساحت این قسمت رو بدست بیار و کامل توضیح بده (شش ضلعی منتظم)
in english

To find the area of the shaded orange region in this regular hexagon, let's analyze the shape step-by-step. Here's a complete breakdown:

### Step 1: Understanding the Structure
- The figure shows a regular hexagon divided into six equilateral triangles.
- In a regular hexagon, if the side length is \( s \), the area of each equilateral triangle is:
  \[
  \text{Area of one triangle} = \frac{\sqrt{3}}{4} s^2
  \]
- Since there are six such triangles, the total area of the hexagon is:
  \[
  \text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4} s^2 = \frac{3\sqrt{3}}{2} s^2
  \]

### Step 2: Identifying the Side Length
- The labeled length "3" represents the side length of the hexagon.

### Step 3: Calculating the Area of the Hexagon
Using \( s = 3 \):
\[
\text{Area of hexagon} = \frac{3\sqrt{3}}{2} \times 3^2 = \frac{3\sqrt{3}}{2} \times 9 = \frac{27\sqrt{3}}{2}
\]

### Step 4: Area of the Shaded Region
The shaded region is one of the six equilateral triangles inside the hexagon. Since each triangle contributes equally to the total area, we can find the area of one triangle by dividing the hexagon’s area by 6:
\[
\text{Area of shaded triangle} = \frac{1}{6} \times \frac{27\sqrt{3}}{2} = \frac{27\sqrt{3}}{12} = \frac{9\sqrt{3}}{4}
\]

### Final Answer
The area of the shaded region is:
\[
\frac{9\sqrt{3}}{4} \approx 3.897
\]

Would you like more details or have any questions about the steps?

### Related Questions
1. What is the perimeter of this hexagon?
2. How does changing the side length affect the area of the shaded region?
3. What is the formula for the area of an equilateral triangle?
4. Can we find the area of a similar shaded region in a pentagon?
5. How would the area change if the hexagon were larger?

### Tip
Remember, a regular hexagon can always be divided into six equilateral triangles, which simplifies area calculations!

مساحت این قسمت رو بدست بیار و کامل توضیح بده (شش ضلعی منتظم)

Find the area of a shaded region in a regular hexagon with a side length of 3 units. This problem covers key geometry concepts, including the area calculation of equilateral triangles and properties of regular hexagons. Suitable for students in Grades 9-10.

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