Can you calculate the area of an irregular object using its parimeter?

Question

Such as a lake on a map

Answer 1

If all you know about the irregular object is the dimension of its perimeter you cannot determine the area. However if you can draw or trace the outline of the object full size or to scale (a known proportion to its full size) you can measure the area with a planimeter instrument just like Answerman said. Land surveyors have used these in the past.
However, if you do not have access to a planimeter you can estimate the area by slicing it with equally spaced horizontal lines. That will give you a bunch of (almost) rectangles having the same height, but different lengths. Add-up the individual rectangular areas. To improve accuracy estimate where to stop each end of each rectangle so that the perimeter is on the center of the rectangle's height at each end.The thinner the slices, the more accurate the estimate.

Answer 2

It's not possible to find the accurate surface area for such complecated objects, And also there is no specific method to do it. But u can do one thing. Get the paper and wrap around the hook. take the area of that paper. Or if u know how to use CAD software then by using that u can easily find the area of any object.

Answer 3

Yes, you can. A device called a planimeter does just this.
The theoretical foundation is Green's Theorem of calculus.
I do not think I can explain it to you here, however.

Essentially you need to trace out the boundary of the region, keeping track of both the instantaneous coordinates of points on the boundary, and the slope of the boundary.