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

To determine the area of the circular field required for a 400-meter race track with 6 lanes, each lane being 1 meter wide, we'll need to break down the problem into several parts.

### 1. Shape of the Track
In standard athletics, a 400-meter race track is not a perfect circle but an **oval** made up of two straight sections (along the straights) and two semi-circular curves at the ends.

- **The total perimeter (circumference) of the track** is 400 meters.
- The track has an **inner edge** for the first lane, and each lane is 1 meter wide.

#### Key Assumptions:
1. **Straight sections**: The straight portions are generally a standard distance.
2. **Curved sections**: The curved sections contribute to the rest of the perimeter and are parts of concentric circles.

For simplicity, let's assume we are working with a **simplified circular approximation** where the track is treated as if it were a perfect circle.

### 2. Perimeter and Radius
The total perimeter of the inner edge (first lane) is 400 meters. The formula for the circumference \(C\) of a circle is:
\[
C = 2\pi r
\]
Where \(r\) is the radius of the inner circle.

Using \(C = 400\) meters, we can solve for \(r\):
\[
r = \frac{400}{2\pi} = \frac{400}{6.28} \approx 63.66 \, \text{meters}
\]
So, the radius of the inner lane is approximately 63.66 meters.

### 3. Width of the Lanes
Since there are 6 lanes, and each lane is 1 meter wide, the outermost lane will have a radius of:
\[
r_{\text{outer}} = 63.66 + 6 = 69.66 \, \text{meters}
\]
Thus, the outer radius of the circular field will be about 69.66 meters.

### 4. Area of the Circular Field
The area \(A\) of a circle is given by the formula:
\[
A = \pi r^2
\]
For the outermost edge of the track (since we are interested in the total area required), we use the outer radius \(r_{\text{outer}} = 69.66\) meters:
\[
A = \pi (69.66)^2 \approx 15247.34 \, \text{square meters}
\]

Thus, the area required for the field, assuming the track is approximately circular, is about **15,247.34 square meters**.

### Conclusion:
The area required for a circular field with a 400-meter race track and 6 lanes of 1 meter width is approximately **15,247.34 square meters**.

Would you like more details or have any questions?

Here are five related questions for deeper exploration:
1. How would the calculation change if the lanes were 1.2 meters wide?
2. What is the area required for a rectangular track with curved ends?
3. How do straight sections affect the overall area calculation?
4. What are the typical dimensions of Olympic-standard tracks?
5. How can we calculate the area for an irregularly shaped track?

**Tip:** For athletics, understanding how tracks are designed (often with elliptical or more complex geometries) can lead to more accurate calculations than a circular approximation!

Discover how to calculate the area needed for a 400-meter race track with 6 lanes, each 1 meter wide. This problem involves circular geometry concepts, including circumference and area formulas, providing an approximate solution of 15,247.34 square meters. Suitable for students in Grades 9-12.

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