Let's say I have 4 points.
- $A(65.3,101.48)$
- $B(61.3,102.52)$
- $C(73.13,102.48)$
- $D(72.96, 101.42)$
There are four lines that connect these points:
- Line 1 is $\overline{AB}$
- Line 2 is $\overline{BC}$
- Line 3 is $\overline{CD}$
- Line 4 is $\overline{DA}$
This would be clearer if you graph the points then connect the A to B to C to D to A. I need to get the area bounded by the 4 lines or inside the 4 points connected.
I thought of using integration and getting the area under the curves but I'm having difficulty with it. Do you know of other ways I can use to get the area?
If this helps, I got the equations of the lines:
- $\overline{AB}: y=-\frac{13}{50}x + \frac{59229}{500}$
- $\overline{BC}: y=-\frac{4}{1183}x + \frac{3038159}{29575}$
- $\overline{CD}: y=\frac{106}{17}x – \frac{300481}{850}$
- $\overline{DA}: y=-\frac{3}{383}x + \frac{1953137}{19150}$
I'd really appreciate the help! I need this for my research paper and I got the graph from my experiment. Thank you in advance!
Edit: From the answers so far, it would seem like geometry/vectors could be used but if someone could give an answer using integration, that would be great because I have to replace 2 of those lines later on with curves.
Best Answer
If you have a simple polygonal closed line with vertices in the points $P_k=(x_k,y_k)$, $k=0,\dots,N$, $P_N=P_0$, ordered counter-clockwise, its area is simply: $$ \frac 1 2 \sum_{k=1}^N (x_{k-1}-x_k)(y_{k-1}+y_k). $$
Just notice that each addend is the area of the trapezoid below the corresponding segment.