Trying to find the square footage of my back yard - know 4 sides

[syano]

I’m trying to figure out the square footage of my back yard without chopping it up into multiple square/triangle combinations…I know the length of the 4 sides… is there a way to find the area of a 4 sided polygon when none of the angles are right angles but you know the length of the 4 sides?

The right side is – 87 feet
The back side is – 84 feet
The left side is – 63 feet
The front side is – 78 feet

If you could solve this or give me the formula it would be much appreciated.

Thank you,

 

[cristo]

What shape is your back yard? Most back yards have two parallel sides-- is this the case with you? If so, you will be able to use quite a simple formula to calculate the area.

 

[neutrino]

Well, you could use integration. Although the principle behind it is chopping up the area into rectangular strips, you actually will not be doing it by hand.

 

[syano]

It is not a parallelogram - thanks though Cristo

Neutrino - Would it be too much to ask to solve this for me using integration? I wouldn't know how to do it other than going outside with some kind of tool to measure distance alone right angles.

 

[HallsofIvy]

You can't do it without know the angles. Imagine a rectangle made of sticks "pinned" together at the corners so the angles can change. By twisting at those pivots you can make the area any number from the maximum value (that of the rectangle) to 0 while having exactly the same side lengths.

 

[syano]

Thanks Hallsofivy... great analogy. I'll try and find a tool to measure these angles.. any recommendations?

 

[lalbatros]

My hint: try to do it algebraïcally.
Try to find out the coordinates of four points A, B, C, D such that the distance match your requirements:

|AB| = 87
|BC| = 84
|CD| = 63
|DA| = 78

Apparently there would be 2x4 = 8 unknowns in this problem.
However, the shape can be displaced or rotated without changing the surface and the distances.
Therefore, there are actually 5 unknowns.
You can position A at the origin and fix one coordinate of another point.

So you can choose for example:

A=(0,0)
B=(0,a) (of course a=87)
C=(b,c)
D=(d,e)

We can then write the equations and solve them for a, b, c, d, e.
By making a drawing with circles around A and B you could visualize the problem and find a way to an easier method of solution.

I solved the problem in Excel using the equation solver and found these coordinates:

A = (0.00 , 0.00)
B = (0.00 , 87.0)
C = (16.0 , 4.50)
D = (61.0 , 48.7)

The surface can be obtained by adding up two traingular surfaces.

I don't know about a simpler method, but maybe there is one ...

 

[Office_Shredder]

Lalbatross, you only know four equations initially, and have five unknowns. One of them gave you your a, so with three equations and four unknowns, how did you do it?

EDIT: Looking at your distance AC, I suspect you don't have it correct, as that is the skinniest backyard ever

 

[Doodle Bob]

Syano,

if you can measure either one of the diagonals of your yard, then that combined with the side measures that you already have would be enough to find the area. Just split the yard into 2 triangles and use Heron's formula on both.

Plus you don't have to worry about accurately computing angles.

 

[syano]

Perfect. Thanks Doodle Bob! That’s a million times easier for me than trying to accurately measure the angles.

Thanks again,

 

[lalbatros]

Lalbatross, you only know four equations initially, and have five unknowns. One of them gave you your a, so with three equations and four unknowns, how did you do it?

I just found one of the infinite number of possible solutions.
Maybe there are other solutions and this would then illustrate that 4 measurements are not enough.
One more measurement in principle should be sufficient to come with one unique solution.

Counting the equations and the unknowns is always useful!

By drawing the segment AB and circles of radii 84 and 78 for BC and DA respectively, it is easy to visualise the undeterminacy.
Indeed, there are many possibilities to join these two circles by a segment of length 63.
Likely all these different solutions have different surfaces.

That's a funny question. It show that triangulation is the most natural method.
Thinking to triangulation also shows that indeed 5 measurements are needed.

 

[lalbatros]

Also interresting is the http://en.wikipedia.org/wiki/Brahmagupta%27s_formula" that I found on Wiki.
(see its extension to non-cyclic quadrilaterals)