Drawing a Circle With a T Square
Drawing a Circle with a Framing Square and 2 Nails
"Squaring the circumvolve" may as still be an unattainable goal for even the best mathematicians, but the November 2012 edition of The Family unit Handyman mag had a tip for how to use a square (of the framing type) and two nails to describe a circle. The author comments, "Don't ask united states why this process works; all nosotros know is that it does." Out of curiosity, I dug out my father'due south erstwhile Audels Carpenters and Builders Guide (printed in 1945) to see if it described the method and if it did, was there an explanation offered. It did, and he did. Read on...
Brand a circle with a foursquare
"Hither'due south a tip for laying out small circles or parts of circles. Tack ii nails to set the bore you want, then rotate a framing square against the nails while y'all hold a pencil in the corner of the square. Yous might need to rub a fiddling wax or some other lubricant on the lesser of the square so it slides hands. Don't inquire us why this process works; all we know is that information technology does. "
They're either very honest or they don't remember the average reader would empathise the explanation.
The Pythagorean theorem is the key, of course, for explaining the reason. For whatsoever right triangle:
a2 + b2 = c2,
where 'a' and 'b' are the lengths of the two perpendicular sides, and 'c' is the length of the hypotenuse. The aforementioned equation besides happens to be (not by coincidence) the equation for a circle of radius 'c,' with the heart at point (0,0). Then, it stands to reason that if all of the parameters are met (three intersecting straight sides with a right angle between two of them), then the locus of points of all permissible value pairs (a,b) will result in a circle. It does not matter whether your value of 'c' represents a radius or a bore. The hypotenuse will always be the length betwixt the two nails and sides 'a' and 'b' will always be the distance between each nail and the 90° vertex. QED
Out of curiosity, I dug out my father's sometime Audels Carpenters and Builders Guide (printed in 1945) to meet if it described the method and if information technology did, was in that location an caption offered. The author did show how to draw a circle with a framing square, and even described how to discover the diameter of a circle whose expanse is equal to the sum of the areas of two given circles (not sure why that would be need past a carpenter). However, an explicit reason for why it all works out is never given.
Here is what is included in the manual:
Outer heel method:
Drive brads at points 50, F, extremities of the given bore. With pencil held at the outer heel M, slide square around with its sides in contact with L, and F, so with the pencil held at Chiliad, describe a semi-circle.
Inner heel method:
Patently if the pencil be held at S, it will be ameliorate guided, than at Thousand. In this method, the distance L'F' should be taken to equal diameter, the inner edges of the square sliding on the tacks-the same edges (in either instance) that guide the pencil.
At the ends of the bore LF (fig. 939) bulldoze brads. Place the outer edges of the square confronting the nails and agree a lead pencil at the outer heel Chiliad, any semi-circle can be described equally indicated.
This is the outer heel method. but a better guide for the pencil is obtained by the inner heel method also shown in the figure.
FIG. 940 - Trouble two: To discover the center of a circle. At the points MS, and LF, where the sides of the square cut the circle when placed in any position with heel in circumference, draw bore and then intersection will be the center of the circle. Why?
FIG. 941 - Problem 3: To draw a circle through three points non in a direct line. Let L, M, and F, be the given points. Join these points with lines LM, and MF, bisecting them at 1 and two. Utilize square with heel at 1 and 2 as shown and the intersection of perpendiculars thus obtained at South, will exist the center of circle which, with radius LS, may be described through LM and F.
To discover the center of a circumvolve.
Lay the square on the circle so that its outer heel lies in the circumference. Marking the intersections of the body and tongue with the circumference. A line connecting these two points is a bore and by drawing another diameter (obtained in the same fashion) the intersection of the two diameters is the eye of the circumvolve every bit shown in fig. 940.
To describe a circle through iii points not in a straight line.
Joint points with straight lines; bifurcate these lines and at the points of bisection erect perpendiculars with the foursquare. The intersection of these perpendiculars is the heart from which a circumvolve may be described through the three points as in fig. 941.
To find the bore of a circle whose surface area is equal to the sum of the areas of two given circles.
Allow O, and H, be the given circles (drawn with diameters LR, and RF at right angles). Suppose diameter of O, be 3 inches, and bore of H, 4 inches. Then points Fifty, F, at these distances from the heel of the foursquare will exist 5 inches apart equally conveniently measured with a two-human foot rule as shown. This altitude LF, or 5 inches, is diameter of the required circle. Proof: LFtwo = LRtwo + RF2, that is 52 = threeii + 4two or 25 = 9+sixteen. (this is as shut as they come to explaining the phenomenon, but not actually).
Posted September one, 2021(original 12/25/2012)
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