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# Advanced Math/square footage of an arced pie shaped lot

Question
QUESTION: Hello Randy,

We have a surveyor's map without the lot size on it. Land Registry can't help us. The surveyor is no longer in business.

front is R 50 with Arc 72.08
back is RAD 233.0 with Arc 335.90
Both sides are 183.0

Can you help?

Lin

ANSWER: I'll be happy to help you but I need a little clarification on the shape. I assume R and RAD are a radii and the Arcs are arc lengths (portions of a circle). Does the shape look like a lozenge with rounded ends? Are the sides parallel and form a rectangle in the middle? Please clarify.

Randy

---------- FOLLOW-UP ----------

QUESTION: Awesome! I tried to attach a .pdf picture of it and couldn't.

Yes, there could be a rectangle in the middle. The property is pie shaped, with the tip of the pie (the front R50) arcing into the lot. Both sides angle outward from the tip of the pie, getting larger as we go further into the property, and the back of the property follows the same arc as the tip of the pie, but is much larger (RAD 233.0)

We are so appreciative of your help!

Lin

OK, I think I got it. It looks like a slice of pie with a small pie shaped portion at the center taken away. I tried to draw it in PowerPoint but crashed and burned. Anyway, you have 2 pie shaped sections, a big one with a radius of 233 ft and a smaller one, whose center is at the same point, with radius 50 ft. Subtracting the area of the smaller section from the area of the larger one will give you the area of your lot.

The length of the arc of the larger section is arc_large =335.9 =  θ･233, so that θ = 1.44 radians (82 degrees). Luckily, this is the same angle you get for the smaller section; arc_small = 72.08 = θ･50  -> θ = 1.44 rads. Thus we have pie shaped sections with the same opening angle and so have the shape described above. Also note that 233 - 50 = 183 = length of sides, so we seem to have a consistent shape.

The area of the pie sections are a fraction of the area of the whole circle with the same radius; the fraction is given by θ/2π= 0.23. Thus the area of the large section is

A_large = (0.23)･(πr^2) = (0.23)･(π･233^2) = 39227 sq ft.

Similarly, the area of the smaller section is

A_small = (0.23)･(π･50^2) = 1806 sq rt.

Thus, you lot size should be

A_large - A_small = 37241 sq ft = 0.86 acres.

I think this works. Good luck!
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment OMG Randy!! Simple words cannot adequately express our appreciation of your help. I was told yesterday by Jonathan at the surveying company who bought out the surveyor who did the original survey in 1969 that they could help me for \$95/hour. When I asked how long it would take, he wasn't very pleasant, stating something like, "Well, they will have to......." I don't think he enjoys his job. It was worth the couple of hours I spent on the internet yesterday trying to find a way to figure it out. I found the AllExperts site! At any rate, you provided not only a quick answer, but in a most pleasant manner. Bless you. You are a gem. Lin

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#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

Education/Credentials
M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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Also an Expert in Oceanography