Quandaries and Queries
Who is asking: Other
Area (S) = 1/2 (r2)theta
where r2 is the radius of the circle squared
where r = radius of the circle and theta is the angle in radians formed by the chord joined at its ends by radial lines.
A friend of mine is a farmer and uses Pivots to irrigate portions of his land. The crop rows are in straight lines that all form chords of a large circle. The intent is to determine area between any two "boundary" rows expressed in acres.
I attempted to apply the above formula using the following numbers
Circle radius = 1325 feet
Using the above formula, the acreage was calculated to be 22.78987374
(43560 square feet = one acre)
Since an annulus of a circle bounded between radii of 1285 and 1325 feet was earlier calculated to be 7.52 acres, the above result seems inconsistent.
Please advise what I need to do to get a good result in case you do not find my answer to be correct.
(note: computing intermediate rows would be accomplished by subtracting the area of the "outer" section from an overall calculation that includes the outer and a larger inner section)
I don't have clear in my mind what you mean by the area between two boundary rows so let me know if I am not finding the correct area. Here is the diagram I have.
M is the midpoint of the chord BA so the angle BMC is a right angle. Let the angle ACB be t radians then |MC|/|CB| = cos(t/2) and hence
Hence the area of the sector CAB is
which is 9.928 acres.
Since triangle BMC is a right triangle, Pythagoras' theorem tells us
Hence the area of the triangle BAC is
which is 9.532 acres.
Thus the area of the region shaded green in the diagram is