## Introduction: Measuring Area of Irregular Shapes

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.

## Step 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

-----------------------------------------------------------------

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.

## Step 2: Triangulating a Complex Shape

To use Heron's formula on a complex shape, all we have to do is split the shape up into a series of triangles, and measure all the edges.

This can be done using Delaunay Triangulation for highest efficiency, but any set of triangles will work. http://en.wikipedia.org/wiki/Delaunay_triangulation

Basically, quadrilaterals are split along their shortest diagonals. This will ensure that your triangles aren't thin and narrow.

Once you have your shape split into triangles and all the edges are measured, calculate Heron's Formula for each triangle, and then add the areas together for a total.

## Step 3: Using a Spreadsheet to Make Life Easier.

Rather than calculate each triangle by hand, you can save yourself some effort by using a spreadsheet. I have attached an example in OpenOffice and Excel formats.

All you have to do is add in values of a,b and c for each triangle, and it will calculate areas and give you a total.*Edit:
I have included versions of the spreadsheet, that have all but the a,b and c cells protected. There is no password, and it's up to individuals if they want the normal or the protected versions. :)*

*To make it a little more clear, the protected versions just prevent you from accidentally erasing or changing the formulas, but they only go up to 30 triangles.*

The normal versions should allow people to adjust the spreadsheet if they want to.

The normal versions should allow people to adjust the spreadsheet if they want to.

*You can thank sonogo for this excellent suggestion.*

## Step 4: Getting Measurements

To get measurements over moderate to large distances, you can use string-lines or measuring (surveyor's) wheels. (I'm a bit surprised I can't find an instructable for how to make one.)

Alternatively, with a bit of practice with a measuring tape, you can learn how to stride regular 1m (**sigh* or for the irredeemably Imperialist, 1 yard*) steps, and pace out rough distances.

The advantages of this method are that:

- there is nothing saying that the triangles have to all lie in the same plane. As long as each triangle section is moderately flat, this method will work with any 3D surface.

- you only need to measure distances. No angles are required. Methods requiring angles can be sensitive to errors of 1/2 a degree over large area. This gives good approximations even if your distances are out by 1 to 2 metres.

If you wanted to measure the surface area of a small 3D shape, you could use string lines to draw triangles over it's surface, and then measure the length of each edge.

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## 18 Comments

For a point-and-click solution, Meander ( http://peacockmedia.co.uk/meander ) allows you to trace the perimeter of any shape you can see on your screen and read off the perimeter and area

Hi shieladixon. Does this measure projected area (I.e. treat the area as a flat plane), or does it take into account changes in elevation?

I come from a farming background in a hilly area. Topo maps and surveyed areas give you the first, but on the ground, a bit of undulation can take a 2ha paddock up to a 2.5ha paddock. Important if you're calculating seed/fertilizer quantities, etc.

Hi, thanks for your interest. At present it allows you to add in a hilliness factor for the line (eg walking route) but doesn't take this into account when calculating the area. Now that you've given me a practical application I'll update the app to do this.

This is OK for very simple approximations, but Green's Theorem and Surface Internals would be far more effective!

Yeah yeah. :D

The thought of keeping track of the line integral x*dy - y*dx as you walk around a paddock sounds reeeeellllllll fun! :P

'nother way is to divide the area into horizontal strips of equal width & measure the lengths of the strip centerlines from edge to edge of shape in question. Add the lengths & multiply by the strip width and you have a pretty good approximation of the area. The narrower the strips, the greater tha accuracy. It's sorta like integration iin calculus if memory serves. This works well for any kind of shape including those with curved edges. If there are holes in the middle, leave them out of the measurements.

This would definitely be a method you'd keep for moderate size areas. :D

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'm afraid my math education is far less comprehensive (and perhaps more ancient) than yours.

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.

The simplest version is the 2/4 rule. Split your area up into an even number of sectors of equal widths. This will give you an odd number of measurements to make.

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.

Perhaps someone could explain?

I thought that you find the area through:

area = (width * height) / 2

Is that any different?

Great job, by the way!

This saves you ensuring that the height is at a perfect right angle. All you need to do is go for a walk around the paddock with a trundle wheel.

Thank you for the information and this method.

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.

Hi sonogo. I've uploaded some password-less, protected versions of the spreadsheet. The issue being that the version I've created has formulas that only go up to 30 triangles. I've left the unprotected versions in as well, if people want to adapt them.

Thank You very much for the very handy and useful method and file you gave in this instructable.

keep the good job.

Thank you for sharing this. Just I know this so useful formula.

I enjoyed reading it. I'm sure I'll be using it some day. Thanks, well done.

This is great! Well written, citations where appropriate, and clear diagrams. And it's got math :-) Too bad you didn't go in for non-Euclidean corrections, but I guess you're not trying to measure the area of a hyperboloid of revolution ;-> Featured and rated!

I used geogebra ( http://www.geogebra.org/cms/ ) for the diagrams, and I'm not entirely certain how to do non-Euclidean corrections. Thank you for the featuring.