Determination of Enclosed Area

[jmas]

Is there any way to calculate area shown in the attached file, mathematically (not numerically and approximately). The restriction two lines are perpendicular and the arc is circular. Any feedback/reference will be highly appreciated.

 

[brandoncdg]

I was curious as to what the problem is trying to solve or why. Sometimes extra information may or may not help to solve a problem. Iterating a solution to who knows what the answer will become and for what reason is not a very good solution for the OP.

Civil Development Group, LLC
Los Angeles Civil Engineering specializing in Hillside Grading

http://civildevelopmentgroup.com

http://civildevelopmentgroup.com/blog

 

[IJR]

Isnt this a triangle plus a circular segment of known length and chord?.

Since the chord length L and arc size S are known, all other dimensions of the arc segment can be found including the rise and the Radius. And once the radius and rise are known area of the circular portion can be determined.

 

[CarlB]

"And once the radius and rise are known area of the circular portion can be determined."

True, but radius & rise cannot be *directly* solved given arc length and chord, as rehashed above.
Or are you Euler's descendant that jmas referred to?

 

[coloeng]

My mathmatician son provided me with the following equation for solving for the radius:

(1/2ie^(-425i/2r))-(1/2ie^(425i/2r))=1/2 chord length

It's much easier to just use an iterative process in a spreadsheet and get as close an approximation as necessary to get an accurate area.

 

[jmas]

To IJR:
Only arc length is known, not chord.

 

[RHTPE]

To jmas - By simple triangles we know the hypotenuse, which in this case is the chord length.


Ralph
Structures Consulting
Northeast USA

 

[TerryScan]

jmas,

I believe there may be a direct solution, but thoughout this thread, I have been unable to determine what are ALL the known values?

You indicate there are two perpendicular lines with known lengths, then you provide a diagram showing a know arc length and radius, but then you say some items are approximate. Then you state only the arc length is known (so, was the radius approximate???)

Please, just state ALL the known values.

 

[RHTPE]

TerryScan - As your can see on the OP's sketch, the boundary consists of 2 (assumed) perpendicular lines plus a (assumed) circular arc segment. So there are 3 known pieces of information:
- The length of the vertical line
- The length of the horizontal line
- The length of the arc segment.
So, given this, the necessary assumptions become:
- The 2 lines are perpendicular
- The arc segment is circular (as opposed to elliptical or free-form).
From the above, it is possible to obtain an exact mathematical solution. The triangular portion is easy. The area of the circular segment become a bit more difficult. But we do know the chord length, so we know that there is but one real arc radius that will give the known arc length and chord length.


Ralph
Structures Consulting
Northeast USA

 

[TerryScan]

I don't have time to noodle through the equations...but here is a site which solves curve data based on only arc and chord length:

http://www.1728.com/circsect.htm