Quandaries
and Queries 

Who is asking: Other Level: All Question: Area (S) = 1/2 (r^{2})theta where r^{2} 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) Thanks, 

Hi Chuck, 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
and hence
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
Harley 

