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Requesting Calculation of Residential Plot area with 4 unequal Lenth

I'm going to get possession of government plot of a housing plot. The total area should be 2160 sq feet equivalent to 3 (three) Katha (a local unit of measurement). The survey drawn a free hand sketch showing 4 length in meters like under
AB = 11.76 meter
BC=15.54 meter
CD = 10.24 meter
DA = 17,76 meter
No angle is marked in the sketch. Now what will be area in square feet or square meter of the plot. I urge those individual expert and expert group to exchange the calculation and help me to produce an area of the aforesaid plot.
Thanks and regards,
Sikder
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Md. Abdur Rashid Sikder
Asked:
Md. Abdur Rashid Sikder
1 Solution
 
Thibault St john Cholmondeley-ffeatherstonehaugh the 2ndCommented:
You will need an angle or a diagonal measurement. With only the lengths of the sides the area plot could be many different sizes.
If you assume it is nearly square then you could have an area of approx 11 x 16 Sq metres (176) but you can't get an accurate value from just that information given.
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aburrCommented:
Thibault... is right.
Consider a rectangle.  IF you push the top side to the right far enough the area will approach zero.
Consider the estimate of the area given you. It is a good one and is about the largest area you can get from those measurements. It is 176 sq meters. If you estimate 3 feet to a meter the estimate is about 1600 sq feet. Much less than the desired 2160 sq feet. You need a larger housing plot.
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dhsindy SparrowRetired considering supplemental income.Commented:
I think the 2160 sqft was an estimate and the area is something less.  The only method I know that could be used to calculate the maximum area would be to draw a diagonal forming two triangles.  Then, write 2 semi-perimeter equations for each triangle forming a function and take the derivative and solve for the maximum area.  I started to do this but it got two involved.  I couldn't find anyway to put the sides together to come closer to 2160 sqft than 2133 sqft.

You would need the survey data to accurately determine the area.  The survey data would show the direction and length of each side.

Using trial-and-error with (2) semi-perimeter equations, I found an approximate maximum value of about 200 sq-meters that is less than 2160 sqft by about 200 sqft.  This approximate maximum was with a diagonal BD of about 20 meters.
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d-glitchCommented:
Angle for maximum areaHere is an Excel spread sheet that will calculate the coordinates of the points and the total area of the plot for the given sides and an arbitrary angle at BAD.  Maximum area is as shown.
     182.03 sq meters ==>  1959.35 sq ft  as dshindy has pointed out.

I have set point A at the origin (0, 0), and the longest segment (AD) as the base.
The coordinates of B (and the height of triangle ABD) are found by trigonometry.
The length of the diagonal BD is from the distance formula.
The area of triangle BCD is found from Heron's formula.
I could stop here, except that I want to make a plot.
Angle ABD is found from the Law of Sines.
Angle BDC is found from the area of the triangle and trig.
Finally the coordinates of C are found from the sum of the angles at D.
Quadrilateral-Calculation-for-ExEx.xlsx
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dhsindy SparrowRetired considering supplemental income.Commented:
I never thought of it before this question but I think any polygon inscribed in a circle with all vertices on the circle is the maximum area.  Seems reasonable since a circle encloses the maximum area for a given perimeter that a cyclic polygon would enclose the maximum area.

Thanks to d-glitch for taking the time to set up the spreadsheet.
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Md. Abdur Rashid SikderCommented:
I am very delighted to find a best solutions made by d-glitch. The Expert did a lot of calculations and applied Excel application to accomplish the assignment. I am happy with the Quadrilateral-Calculation-for-ExEx.xlsx and details of the deliberations.
Thanks expert of d-glitch for his untiring endeavor.
Best wishes,
Sikder
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dhsindy SparrowRetired considering supplemental income.Commented:
Just note that noone found the correct answer because there is not enough information.

The answer given is the maximum possible area.  To get your correct answer we need an angle or diagonal information.
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aburrCommented:
The asker in ID: 41806096 essentially awarded d-glitch the points. In my mind Thibault... first pointed out the inadequacy of the data furnished so he should get a few points.
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dhsindy SparrowRetired considering supplemental income.Commented:
I agree with "aburr".  Even finding a maximum requires that you assume the quadrilateral is a cyclic quadrilateral.
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