Calculating Area Around a Path

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In summary, the conversation discusses finding the total fertile area of a river given its coordinates and the distance at which the soil is fertile. The current method of obtaining the area, by multiplying the river's length by the distance, is inaccurate for small angles between lines. The suggested solution is to break the area into smaller rectangles and subtract the overlapping triangular areas. The speaker also mentions the need to incorporate angles and create an algorithm for automatically calculating the area for different arrangements of coordinates.
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rowardHoark
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Say "A" is a river. It is a linear approximation because only several coordinate points are known. The total length is "l" (obtained from the coordinates), "a" represents the distance from the river at which the soil is fertile.

The goal is to obtain the total fertile area. If [itex]Area=l*a[/itex] the result is very inaccurate for small angles between lines, as an extra, overlapping, area (represented as red in "B") is added.

How could I obtain a more accurate value?
 
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  • #2
Well, you know how to find the area of triangles, don't you? Your area can be reduced to a series of rectantgles with the area of triangles, where they overlap, subtracted off.
 
  • #3
Thank you, HallsofIvy. I will try to incorporate angles into my mathematical model. I know the coordinates of the points where the lines meet, hence can derive angles.

The system is dynamic. The coordinate points are generated randomly, therefore I need to be able to know the exact area for various arrangements, and it has to be done automatically using an algorithm.
 

Related to Calculating Area Around a Path

1. How do you calculate the area around a path?

To calculate the area around a path, you can use the formula A = ∫y dx, where y represents the function of the path and dx represents the infinitesimal change in x. This formula is also known as the definite integral of the function y with respect to x.

2. What tools or techniques are used to calculate the area around a path?

There are various tools and techniques that can be used to calculate the area around a path, such as integration, calculus, and geometry. These methods involve breaking down the path into smaller segments, finding the area of each segment, and then summing them up to get the total area. Software programs like MATLAB and Wolfram Alpha can also be used to calculate the area around a path.

3. Can the area around a path be calculated for any shape or curve?

Yes, the area around a path can be calculated for any shape or curve as long as the path can be represented by a function. This means that the path must have a continuous and well-defined mathematical equation that can be integrated to find the area.

4. How is the area around a path related to the perimeter of the path?

The area around a path and the perimeter of the path are related but they are not the same. The perimeter of a path is the distance around the path, while the area around a path is the space enclosed by the path. In some cases, the perimeter can be used to estimate the area around a path, but it is not always accurate.

5. What are some real-world applications of calculating the area around a path?

Calculating the area around a path has various real-world applications, such as in engineering, architecture, and physics. It can be used to determine the amount of material needed to build structures, the amount of paint needed to cover a curved surface, and the amount of fluid or air that can pass through a particular path. It is also used in GPS and navigation systems to calculate the area of landmasses and bodies of water.

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