Determination of Enclosed Area
Determination of Enclosed Area
(OP)
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.





RE: Determination of Enclosed Area
Civil Development Group, LLC
Los Angeles Civil Engineering specializing in Hillside Grading
http://civildevelopmentgroup.com
RE: Determination of Enclosed Area
1- two perpendicular lines with 210 and 260 m length.
2- a circular arc which connect end of these lines together.
The question: Is there any formula which give us enclosed area in terms of before mentioned data.
Regards,
RE: Determination of Enclosed Area
Civil Development Group, LLC
Los Angeles Civil Engineering specializing in Hillside Grading
http://civildevelopmentgroup.com
http://civildevelopmentgroup.com/blog
RE: Determination of Enclosed Area
RE: Determination of Enclosed Area
RE: Determination of Enclosed Area
There is a device called a Planimeter that, when calibrated, can give you the area.
Knowing a bit more about the arc (like chord height & radius) can get you to a formula for the circular section. That, combined with the area for the triangle, will give you the total area.
Otherwise, it's an iterative process in CAD to draw an arc of matching length that connects the ends of the 2 perpendicular lines.
Ralph
Structures Consulting
Northeast USA
RE: Determination of Enclosed Area
You can put the 2 formulas in Excel & iterate.
I get a radius of about 181.36.
RE: Determination of Enclosed Area
h=r-(r^2-l^2/4)^.5 If angle angle between radii at ends of arc is not more than 180 degrees.
h=r+(r^2-l^2/4)^.5 if angle is more than 180 degrees.
You are given the chord length "l" and the rise "h" value being the distance between midpoints of arc and chord.
Calculate radius "r".
Assuming the drawing is in ACAD, draw two tangents at end points or any where on the arc and their normals at point of tangencies. Intersection of the normals (or perpendiculars) will be the radius of the arc.
RE: Determination of Enclosed Area
Suppose length of circular arc is L and half of distance between end of lines is b (see attached file). then we will have:
RxAlpha=L/2
Rxsin(Alpha)=b
So we have:
Alpha-Cxsin(Alpha)=0
where C=L/2/b
This will give us a non-linear equation that i don't think whether it has a straight solution.
RE: Determination of Enclosed Area
Although some responders above indicated otherwise, maybe they can enlighten us :)
Some formulas at-
http://mathworld.wolfram.com/CircularSegment.html
RE: Determination of Enclosed Area
CarlB (& everyone else) - There IS a direct solution that involves some equation manipulation that eludes me right now. It would have to be in 3 forms, one for an arc angle less than 180°, one for a half circle (arc angle = 180°), and another for an arc angle greater than 180°.
See page 6-18 in the 8th Edition AISC Manual. I have been spoiled by 25 years of AutoCAD, so my geometric analysis skills are very rusty.
Ralph
Structures Consulting
Northeast USA
RE: Determination of Enclosed Area
That's just not right, saying you know there is a direct solution then not providing it.
People much smarter than I have said there is not an exact solution, so if you can find one you would rock the mathematical world I think.
Here's a power series solution to thetranscendental equations, whatever that means :)
http://an
RE: Determination of Enclosed Area
Sorry CarlB. From the equations in the AISC Manual, and knowing these 3 things: Arc length, chord length, and that it is a circular arc, intuitively I know there is but one radius. Knowing the radius leads to the complete solution that the OP wants. Finding the solution intrigues me, and when I get some work completed that actually pays the bills, I'll return back to this challenge.
Ralph
Structures Consulting
Northeast USA
RE: Determination of Enclosed Area
Civil Development Group, LLC
Los Angeles Civil Engineering specializing in Hillside Grading
http://civildevelopmentgroup.com
http://civildevelopmentgroup.com/blog
RE: Determination of Enclosed Area
brandoncdg -
Why does it matter? This is likely a lot area question. The OP wants input regarding a direct mathematical solution to the question. Those of us well-versed in AutoCAD know that we can enclosed the area with a polyline and find the area.
But there is no way in AutoCAD to draw an arc knowing just the chord & arc lengths, so it becomes an iterative process. Knowing these 3 things: Arc length, chord length, and that it is a circular arc can lead to only one radius - once that is solved, the rest is easy.
Ralph
Structures Consulting
Northeast USA
RE: Determination of Enclosed Area
RE: Determination of Enclosed Area
chicopee - If you study the equations I referenced in AISC's Eigth Edition Steel Construction Manual, and if you carefully consider that given a circular arc and knowing its chord length and arc length, there is but one radius for the arc. Once the radius has been determined, the rest is easy.
Ralph
Structures Consulting
Northeast USA
RE: Determination of Enclosed Area
RE: Determination of Enclosed Area
??? Of course there is an exact solution. It may not be expressed in a single function, and may involve irrational and trancendtal numbers (square roots and pi), but the algorithms given above will find a single solution, to any degree of precision you wish to calculate.
The closest to a closed form solution, applicable to your diagram, where I replace your numbers by variables:
x (lateral distance=210 in your fig)
y (vertical distance= 260 in your fig)
s (arc length = 425 in your fig)
a = chord length = sqrt(x^2 + y^2)
unknowns:
A = angle of the arc "s" (in radians)
R = radius of the arc "s"
equations:
s = R*A
a = 2*R*sin(A/2)
substituting,
a = (2s/A)*sin(A/2)
this last equation must be solved, most readily by iteration, to find the angle A, knowing a and s. There is only one exact angle "A", between 0 and 2*pi, that will solve the equation, and it can be determined "exactly" for any given value of exactly. Once A is known, the radius R can be calculated, and the center of the arc "s" located. Calculation of the area follows by calculation of the triangles and the circular segment thus formed.
RE: Determination of Enclosed Area
Civil Development Group, LLC
Los Angeles Civil Engineering specializing in Hillside Grading
http://civildevelopmentgroup.com
http://civildevelopmentgroup.com/blog
RE: Determination of Enclosed Area
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.
RE: Determination of Enclosed Area
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? :)
RE: Determination of Enclosed Area
(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.
RE: Determination of Enclosed Area
Only arc length is known, not chord.
RE: Determination of Enclosed Area
To jmas - By simple triangles we know the hypotenuse, which in this case is the chord length.
Ralph
Structures Consulting
Northeast USA
RE: Determination of Enclosed Area
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.
RE: Determination of Enclosed Area
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
RE: Determination of Enclosed Area
http://www.1728.com/circsect.htm