A man is running around a regular hexagonal field of side length 6 m, so that he always at a distance 3 metre from the nearest boundary. Find the length of path travelled by him in one round

The man is running around a hexagonal field, staying at a constant distance of 3 meters from the nearest boundary. This situation suggests that he is following a path parallel to the boundary of the hexagonal field, which forms a smaller, inner hexagon at a distance of 3 meters from the original boundary.

### Step-by-step solution:

1. **Original Hexagon Side Length**:  
   The side length of the original hexagonal field is 6 meters.

2. **Inner Hexagon Side Length**:  
   Since the man maintains a distance of 3 meters from the boundary, the side length of the inner hexagon is reduced by 3 meters on each side. Therefore, the side length of the inner hexagon is:
   \[
   \text{Inner Hexagon Side Length} = 6 - 3 = 3 \, \text{meters}.
   \]

3. **Formula for Perimeter of a Regular Hexagon**:  
   The perimeter \( P \) of a regular hexagon is given by:
   \[
   P = 6 \times \text{side length}.
   \]
   For the inner hexagon, the side length is 3 meters.

4. **Perimeter of the Inner Hexagon**:  
   The perimeter of the inner hexagon is:
   \[
   P = 6 \times 3 = 18 \, \text{meters}.
   \]

Thus, the length of the path travelled by the man in one round is **18 meters**.

Would you like more details or clarifications? Here are some related questions to further your understanding:

1. What is the formula for the perimeter of any regular polygon?
2. How do we calculate the area of a regular hexagon given the side length?
3. If the man ran at a constant speed, how would we determine the time taken for one round?
4. What if the distance from the boundary was 2 meters instead of 3? How would the solution change?
5. How does the perimeter of the hexagon change if the side length is doubled?

**Tip**: For any regular polygon, the perimeter is always the number of sides multiplied by the side length!

A man is running around a regular hexagonal field of side length 6 m, so that he always at a distance 3 metre from the nearest boundary. Find the length of path travelled by him in one round.

Discover how to calculate the path length of a man running around a regular hexagonal field with a side length of 6 meters, staying 3 meters away from the nearest boundary. This geometry problem involves understanding the perimeter of regular polygons and applying the formula for hexagons. Ideal for students in Grades 8-10.

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