This instructable combines a few mathematical tricks to enable you to calculate the area of irregular shapes.
I come from a farming background. One of the things we often had to do, was measure the area of a section of land in order to calculate how much fertiliser was required.
This can be done from maps, (E.g. using Google Maps measure tools) but this doesn't take into account variations in height. Our farm was in a moderately hilly area, and often the real area of a paddock was 20% more than what the maps said.
This instructable could also be used to measure area of irregular plane shapes, as well as surface area of 3D shapes.
I come from a farming background. One of the things we often had to do, was measure the area of a section of land in order to calculate how much fertiliser was required.
This can be done from maps, (E.g. using Google Maps measure tools) but this doesn't take into account variations in height. Our farm was in a moderately hilly area, and often the real area of a paddock was 20% more than what the maps said.
This instructable could also be used to measure area of irregular plane shapes, as well as surface area of 3D shapes.
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Signing UpStep 1: Heron's Formula - Calculating the Area of a Single Triangle.
Heron's formula is a way to calculate the area of ANY triangle, knowing the lengths of it's three sides, a, b and c. More information can be found at http://en.wikipedia.org/wiki/Heron%27s_formula
Firstly, calculate the semiperimeter. This is just half of the triangle's perimeter.
s = (a + b + c) / 2
In our example,
s = (6.56 + 6.01 + 5.76)/2
s = 9.165
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Then, calculate the following differences:
s - a
s - b
s - c
E.g.
s - a = 9.165 - 6.56 = 2.605
s - b = 9.165 - 6.01 = 3.155
s - c = 9.165 - 5.76 = 3.405
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Finally, substitute your answers into the following equation:
A = sqrt( s * (s-a) * (s-b) * (s-c) )
(Multiply the three differences together, along with the semiperimeter, and then take the square root.)
A = sqrt( 9.165 * 2.605 * 3.155 * 3.405)
A = sqrt( 256.482)
A = 16.015
This gives the area of a single triangle.
Firstly, calculate the semiperimeter. This is just half of the triangle's perimeter.
s = (a + b + c) / 2
In our example,
s = (6.56 + 6.01 + 5.76)/2
s = 9.165
-----------------------------------------------------------------
Then, calculate the following differences:
s - a
s - b
s - c
E.g.
s - a = 9.165 - 6.56 = 2.605
s - b = 9.165 - 6.01 = 3.155
s - c = 9.165 - 5.76 = 3.405
------------------------------------------------------------------
Finally, substitute your answers into the following equation:
A = sqrt( s * (s-a) * (s-b) * (s-c) )
(Multiply the three differences together, along with the semiperimeter, and then take the square root.)
A = sqrt( 9.165 * 2.605 * 3.155 * 3.405)
A = sqrt( 256.482)
A = 16.015
This gives the area of a single triangle.
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The thought of keeping track of the line integral x*dy - y*dx as you walk around a paddock sounds reeeeellllllll fun! :P
I'd imagine this would take 4 hours on a large paddock, which tend to be straight edged any case. ;)
In the situations where this is necessary (E.g. measuring a curved lawn), the various forms of Simpson's Method may be a more accurate form of this approach. It gives approximately twice as much accuracy, allowing you to make the strips twice as big.
For gardens/areas that are almost circular, using Simpson's Method from the centre to measure out arcs (Keep the width of the ends constant) and divide the total by two is also a good approach.
I looked up Simpson's rule and it made my head hurt. Didn't do very well in the calculus classes I attempted because of those lousey headaches.
My thoughts run more along the lines of the rectangular method using midpoint approximation. I often have to estimate irregular shapes on plans measuring from a few hundred square feet to hundreds of acres. After using this method for a number of years, I find that if I choose the width of the rectangles correctly, my results are close enough the output of high end digitizers or autocad and don't take much time to do using a simple measuring wheel and a straight edge. If the margins are made up of straight lines, subdividing into simple straight sided shapes can be a quicker approach if the original shape is regular enough but that's a judgement call.
I'm sure that the math approach will provide better accuracy if you have the right information but I don't find the error in the approach that I use to be significant in my world.
E.g. assume we split the area into six sectors, and seven measurements, which are [1,2,1,3,1,2,2].
Take the first and last value and add them together.
1 + 2 = 3
Now take each even value, add them together and multiply by four.
2 + 3 + 2 = 7
7 * 4 = 28
Take the remaining odd values, add them together and multiply by two.
1 + 1 = 2
2 * 2 = 4
Add all these results together and divide by 3.
3 + 28 + 4 = 35
35 / 3 = 11 2/3
Simpson's rule is based on taking polynomial approximations of each section.
I thought that you find the area through:
area = (width * height) / 2
Is that any different?
Great job, by the way!
I would like to comment on the xls file you made. it is okay but I think you should have put some protection on the cells other than the value of a,b,and c
if some one write on the cell of S or a-S for example it would damage the file. some people doesn't know how to drag the formula to the other cells.
otherwise it is a great job you've done in there..
thanks.
keep the good job.