Area of a general n-sided polygon

[I_am_learning]

Finding the area of an irregular polygon with n side is quite easy when we are given the length of all of the n sides and the length of (n-3) specific diagonals. This way, we get (n-2) triangles whose areas can be calculated using Heron's formula and then added up.

But what if the length of the (n-3) diagonals provided doesn't make (n-2) triangles, such as this case:

The polygon is still fully determined by the given measurements, but calculating the area is difficult.
Is there some sort of generic formula for such cases? Like maybe using matrices. :D

I thought of making a mobile application to help real-estates peoples calculate the area of lands, and came-up with this question.
Thank you.

 

[SteamKing]

There are several different methods of finding areas of general polygons:

http://en.wikipedia.org/wiki/Polygon

When you say 'real estate peoples', are you referring to land surveyors or someone else?

 

[I_am_learning]

There are several different methods of finding areas of general polygons:

http://en.wikipedia.org/wiki/Polygon

When you say 'real estate peoples', are you referring to land surveyors or someone else?

There is no formula for finding area when n sides and n-3 diagonals are known.
The surveyors formula seems to be the best way to go. So from the given information, I should somehow find the coordinates of all the vertices and also sort them counter clockwise or clockwise.

By real state peoples I just meant anyone who is involved in buying or selling of lands.

 

[SteamKing]

Well, in the US, the land surveyor is the professional who confirms and measures the boundaries of a particular plot of land. There is usually a legal description of the land produced for a deed of title to the land, which would contain the area enclosed by these boundaries.

The proper orientation for your vertex coordinates is counterclockwise to calculate the positive area of the figure. If you use a clockwise orientation, the result will be a negative area. This comes in handy if you want to evaluate the area of complex, non-convex polygons, say a polygon with a hole in it.

 

[CellCoree]

I would like to first commend you on your idea for a mobile application to help real estate professionals calculate the area of lands. It is important to have accurate measurements in this field, and your idea could definitely be helpful.

In regards to your question about a generic formula for calculating the area of a general n-sided polygon, unfortunately, there is no one formula that can be applied in all cases. The formula you mentioned, using Heron's formula and dividing the polygon into (n-2) triangles, is a commonly used method for finding the area of irregular polygons. However, as you mentioned, it may not always work if the given diagonals do not form (n-2) triangles.

In such cases, one approach could be to use trigonometry and break the polygon into smaller, simpler shapes such as triangles, rectangles, and parallelograms, and then calculate their individual areas and add them up. Another approach could involve using coordinate geometry and dividing the polygon into smaller polygons with known area formulas, and then adding them up.

Alternatively, as you suggested, using matrices could also be a possible approach. This would involve representing the polygon as a matrix and using mathematical operations to calculate its area. However, this may be a more complex and time-consuming method compared to some of the other techniques mentioned above.

In conclusion, there is no one generic formula for calculating the area of a general n-sided polygon. It depends on the given measurements and the shape of the polygon. I would suggest exploring different methods and techniques, and using the one that is most suitable for the given polygon. Best of luck with your mobile application!