![]() ![]() Angle ∠AOF is our second approximation of a degree.Let "F" be the new point that is one-third of the way from C to E.Use mweiss' method to approximately trisect angle ∠COE.Angle ∠COE is slightly more than 0.0003°.to make an angle ∠AOE that exceeds 1° by slightly more than 0.0002°.If that accuracy is not good enough for you, you are welcome to repeat the process: On a (huge!) one meter diameter protractor, that is a worst case error of less than 4 microns, or one-seventh of a thousandth of an inch.įormulas used to calculate the "shy-ness" of the outer angles: In mweiss' method, the outer angles ∠AOC and ∠DOB are shy of 1° by slightly more than 0.0001°, and the inner angle ∠COD exceeds 1° by slightly more than 0.0002°. Steps 5 and 8 are a way to wrap a line segment into a circle. Step 6 is a way to straighten out the circle back to a line segment. Note that steps 1-5 are equivalent to using a compass to draw a circle on a paper. Step 9: Optionally, secure the string in place. Step 8: Put the marked-off string back in the groove. The number of parallel lines needs to be one more than the number of increments.) (You can do this by making a set of evenly-spaced parallel lines, and placing the taut string between points on the far parallel lines. ![]() Step 7: Measure off however many even increments you want along the string. Step 6: Cut off exactly one lap of the string. Step 5: Wrap the string around the cylinder (in the groove). Step 4: Make a fine groove in the cylinder, of even depth and non-wavering axial elevation. Step 3: Make a cylinder out of the machinable material (perhaps using a lathe). Find a string material that is reasonably non-stretchy. Step 1: Find a machinable material that is reasonably incompressible. Given the imprecision involved in using mechanical construction tools (how thick is the tip of your pencil? how smoothly can you draw an arc with a compass? how 'straight' is your straightedge?), and the inherent limits involved in reading or using a protractor (can you even measure a degree to less than 0.1 degree precision with a protractor anyway?), this would seem to be good enough for almost all conceivable purposes. The resulting angles $\angle AOC, \angle COD, \angle DOB$ are not exactly 1 degree each, but the difference between the actual measures and the desired measures are less than 1 part in 10,000. Join $A$ to $B$ to create segment $\overline$.Let $O$ be the center of a circle, and let $A, B$ be points on the circle such that arc $AB$ measures 3 degrees.However, the following incorrect trisection method produces angles that are very, very close to correct: In principle, it is not possible to trisect a $3^\circ$ angle using only a compass and straightedge. As Will Orrick says in the comments under user20315's answer, it is possible, with straightedge and compass, to construct a regular 120-gon, and therefore it is possible to mark off every 3 degrees on a circle.Ĭan we get any farther? It depends on how much precision you require, and how much error you are willing to tolerate. ![]()
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