# Area of a general n-sided polygon

• I_am_learning
In summary, there are various methods for finding the area of irregular polygons, including using Heron's formula for (n-2) triangles and using matrices. However, if the given measurements do not result in (n-2) triangles, it can be difficult to calculate the area. Land surveyors are professionals who are responsible for determining the boundaries and area of a plot of land, and their formula is often the best approach. It is also important to determine the proper orientation of the vertex coordinates, typically counterclockwise, in order to accurately calculate the area.
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.

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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?

SteamKing said:
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.

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.

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!

## 1. What is the formula for finding the area of a general n-sided polygon?

The formula for finding the area of a general n-sided polygon is A = (1/2) * n * s * h, where n is the number of sides, s is the length of a side, and h is the height of the polygon.

## 2. How do you calculate the height of a general n-sided polygon?

To calculate the height of a general n-sided polygon, you can use the formula h = (s/2) * cot(180/n), where s is the length of a side and n is the number of sides. Alternatively, you can use trigonometry to find the height by dividing the polygon into triangles and using the Pythagorean theorem.

## 3. Can the area of a general n-sided polygon be negative?

No, the area of a polygon cannot be negative. It represents the amount of space enclosed by the polygon and therefore must be a positive value.

## 4. How is the area of a general n-sided polygon related to its perimeter?

The area of a general n-sided polygon is not directly related to its perimeter. However, as the number of sides increases, the perimeter also increases, and therefore the area also increases.

## 5. Can the area of a general n-sided polygon be calculated without knowing the length of a side?

Yes, the area of a general n-sided polygon can be calculated without knowing the length of a side by using the formula A = (1/2) * n * a * h, where a is the apothem (the distance from the center of the polygon to the midpoint of a side) and h is the height. Alternatively, the area can be calculated by dividing the polygon into triangles and using the formula A = (1/2) * base * height for each triangle.

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