for conducting a race of 400 m and for having 6 lanes of width 1m the area of the circular field required will be

To determine the area of a circular field required for a 400-meter race with 6 lanes, each of width 1 meter, we need to calculate the area of the field including the track and the inner circular area.

### Understanding the Problem

1. **Track Length:** The total length of the race is 400 meters. In a circular track, this is the perimeter of the innermost lane.
2. **Lanes:** There are 6 lanes, each with a width of 1 meter.
3. **Radius of the Inner Lane (r):** Let's assume the radius of the innermost lane is \( r \).

#### Step-by-Step Solution

1. **Perimeter of the Innermost Lane:**

   The perimeter (circumference) of the innermost lane is 400 meters:
   \[
   2 \pi r = 400 \implies r = \frac{400}{2 \pi} = \frac{200}{\pi} \, \text{meters}
   \]

2. **Outer Radius of the Track:**

   Since there are 6 lanes, each 1 meter wide, the outermost lane will be at a distance of 6 meters from the innermost lane. Therefore, the outer radius \( R \) is:
   \[
   R = r + 6 = \frac{200}{\pi} + 6 \, \text{meters}
   \]

3. **Area of the Circular Field:**

   The required area is the area of the outer circle with radius \( R \):
   \[
   \text{Area} = \pi R^2 = \pi \left( \frac{200}{\pi} + 6 \right)^2
   \]

Let's compute this area.The area of the circular field required for a 400-meter race with 6 lanes, each 1 meter wide, is approximately **15,245.49 square meters**.

Would you like more details on this calculation or have any other questions?

### Related Questions:
1. How do different lane widths affect the total area of a track?
2. What is the formula for calculating the area of a circular track with multiple lanes?
3. How can the length of each lane be determined in a circular race track?
4. How does the shape of the track affect the running dynamics of the race?
5. What are the standard dimensions for athletic tracks in international competitions?

**Tip:** When designing a race track, remember to consider not only the total area but also the safety zones and space for spectators.

Learn how to calculate the area required for a circular field to conduct a 400-meter race with 6 lanes, each 1 meter wide. This solution involves geometry concepts like circumference and area of circles, suitable for students in Grades 9-12.

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