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.