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Is there a formula to find out the perimeter of an irregular shape on lattice points when you already know the area?

I am asking this as I have a test tomorrow, and one of the questions that could come up is the area/perimeter of an irregular shape on lattice points. I already know how to find out the area using Pick's theorem, however there doesn't seem to be anything on the internet which talks about finding out the area by a formula and when you already know the area. This website could help though. So, is there a formula to find out the perimeter of an irregular shape on lattice points when you already know the area?

Also, is there a way to find out the perimeter of an irregular shape when you already know the area, but the shape isn't on lattice points? Thanks!

Irregular shape on lattice points

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    $\begingroup$ If the shape is irregular, but all corners are on lattice points, then using the Pythagorean theorem repeatedly is, I think, the best way to calculate the perimeter. $\endgroup$ – Arthur Jul 26 '17 at 12:01
  • $\begingroup$ You take the area, write it down, then turn the page and add the distances between consecutive vertices. $\endgroup$ – Nina Simone Jul 26 '17 at 12:01
  • $\begingroup$ @Arthur- Use Pythagorean Theorem repeatedly on what? $\endgroup$ – bio Jul 26 '17 at 12:32
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Use the distance formula repeatedly for each side.

Distance Formula

Distance between two points whose coordinates are $(x_1 , y_1)$ and $(y_1 , y_2)$ is given by the formula $$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} $$

This can be proved using the Pythagoras theorem.

For simplicity, you can set the origin at one end of the sides to simplify the formula each time.

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