Finding the area of my lake Last year I took AP Calculus BC and learned some basic information on areas of irregular objects (Riemann sums and thats about it). I thought it was very interesting and this year we were assigned a math project on the topic of our choice.
I have a lake behind my house that I decided to find the area of for the project. I went into google maps and clicked on some points, but I'm not sure what to do from here. I've been reading online and saw some interesting methods like Green's theorem and Planimetry, but I'm not sure how to utilize those in the context of my lake since I don't have an equation. I assumed I would have to plot these points on a graph to start off with, but I'm not sure where or how to do that using the information from my Google Map. Can anyone help me mathematically go about this? For reference, an image of the lake is attached below:
(I know it shows the area, I just want to learn - and show - how to do that)
Heres a link to the lake for anyone that wants to experiment with the 'points' on a google map. You just right click and press 'measure distance'.
https://goo.gl/maps/M8WH6wPYGyxMP97z9
 A: Something that might interest you is the shoelace formula for the area of a polygon given the cartesian coordinates of the vertices. According to the wikipedia page:

[The shoelace formula] can be verified by dividing the polygon into triangles, and can be considered to be a special case of Green's theorem.

Mathologer has a youtube video on this topic. His videos are a delight to watch and I always learn something new when watching them, so I recommend checking that out.
As for applying the formula to your situation, we would like to choose an origin and set up a cartesian coordinate system. One way to do this is using a program like photoshop which displays the cartesian coordinates of the cursor in terms of pixels. Then using the shoelace formula you could find an estimate $A$ of the area of the lake in square pixels. After that you could convert pixels to metres by having a reference length in the image (e.g. by using the length scale provided by google) and measuring that in terms of pixels again using a photo editor. From this we could find the value of $x$ in the conversion $1 \ \text{pixel} = x \ \text{metres}$, then our estimate of the area of the lake in square metres is $x^2 A$.
As pointed out in the comment section of the question, the calculation of the area won't be exact, so ideally we would want to add an uncertainty $\delta$ in the measurement of the area of the lake so that we are confident that the true area of the lake in square metres lies in the interval $[x^2 A - \delta, x^2 A + \delta]$.
There are various ways to do this, one being to find the area of a polygon that envelopes the lake and finding the area of a polygon lying inside the lake. These will give you upper and lower bounds for the true area, allowing for a suitable choice of $\delta$.
