Draw a Circle With a Carpenters Square
Cartoon a Circumvolve with a Framing Square and 2 Nails
"Squaring the circumvolve" may as however exist an unattainable goal for fifty-fifty the all-time mathematicians, but the November 2012 edition of The Family Handyman magazine had a tip for how to use a square (of the framing type) and two nails to describe a circumvolve. The author comments, "Don't ask us why this process works; all we know is that it does." Out of curiosity, I dug out my father'southward old Audels Carpenters and Builders Guide (printed in 1945) to run into if information technology described the method and if it did, was there an explanation offered. It did, and he did. Read on...
Make a circle with a foursquare
"Hither's a tip for laying out small circles or parts of circles. Tack two nails to gear up the diameter you desire, then rotate a framing square against the nails while you agree a pencil in the corner of the foursquare. Y'all might need to rub a little wax or some other lubricant on the bottom of the square then it slides easily. Don't enquire u.s. why this process works; all we know is that it does. "
They're either very honest or they don't think the average reader would empathize the caption.
The Pythagorean theorem is the central, of class, for explaining the reason. For any right triangle:
a2 + btwo = c2,
where 'a' and 'b' are the lengths of the two perpendicular sides, and 'c' is the length of the hypotenuse. The aforementioned equation also happens to be (not by coincidence) the equation for a circle of radius 'c,' with the eye at point (0,0). And so, it stands to reason that if all of the parameters are met (3 intersecting directly sides with a correct angle betwixt 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 diameter. The hypotenuse volition always be the length between the two nails and sides 'a' and 'b' volition always be the distance between each boom and the 90° vertex. QED
Out of marvel, I dug out my male parent's old Audels Carpenters and Builders Guide (printed in 1945) to run across if it described the method and if it did, was at that place an explanation offered. The author did show how to draw a circumvolve with a framing square, and fifty-fifty described how to find the diameter of a circle whose expanse is equal to the sum of the areas of ii given circles (not sure why that would be need past a carpenter). Even so, 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 Grand, slide square effectually with its sides in contact with L, and F, then with the pencil held at M, describe a semi-circumvolve.
Inner heel method:
Apparently if the pencil be held at S, it will be better guided, than at M. In this method, the distance 50'F' should be taken to equal diameter, the inner edges of the square sliding on the tacks-the same edges (in either case) that guide the pencil.
At the ends of the bore LF (fig. 939) drive brads. Place the outer edges of the foursquare against the nails and concur a lead pencil at the outer heel G, any semi-circle can be described as indicated.
This is the outer heel method. simply a ameliorate guide for the pencil is obtained by the inner heel method also shown in the figure.
FIG. 940 - Problem two: To observe the middle of a circle. At the points MS, and LF, where the sides of the square cutting the circle when placed in whatever position with heel in circumference, draw bore and and then intersection will exist the heart of the circle. Why?
FIG. 941 - Problem 3: To describe a circle through 3 points non in a straight line. Let 50, K, and F, be the given points. Join these points with lines LM, and MF, bisecting them at one and 2. Utilise square with heel at 1 and ii equally shown and the intersection of perpendiculars thus obtained at S, volition be the heart of circumvolve which, with radius LS, may be described through LM and F.
To find the middle of a circle.
Lay the square on the circle so that its outer heel lies in the circumference. Marking the intersections of the torso and tongue with the circumference. A line connecting these two points is a bore and past drawing another diameter (obtained in the same manner) the intersection of the 2 diameters is the heart of the circumvolve equally shown in fig. 940.
To describe a circle through three points not in a straight line.
Joint points with directly lines; bisect these lines and at the points of bisection cock perpendiculars with the square. The intersection of these perpendiculars is the eye from which a circumvolve may be described through the three points as in fig. 941.
To find the diameter of a circle whose area is equal to the sum of the areas of two given circles.
Let O, and H, be the given circles (drawn with diameters LR, and RF at correct angles). Suppose bore of O, exist iii inches, and diameter of H, four inches. Then points 50, F, at these distances from the heel of the square will be 5 inches apart as conveniently measured with a 2-foot rule as shown. This distance LF, or 5 inches, is bore of the required circle. Proof: LF2 = LR2 + RFii, that is 52 = 32 + 42 or 25 = 9+sixteen. (this is as close equally they come to explaining the phenomenon, simply not actually).
Posted September i, 2021(original 12/25/2012)
Source: https://www.rfcafe.com/miscellany/smorgasbord/drawing-circle-with-framing-square-and-2-nails.htm
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