How to Draw a Circle in a Square
Drawing a Circle with a Framing Square and 2 Nails
"Squaring the circle" may as however be an unattainable goal for even the best mathematicians, just the Nov 2012 edition of The Family Handyman magazine had a tip for how to use a foursquare (of the framing type) and two nails to draw a circle. The author comments, "Don't ask usa why this process works; all we know is that it does." Out of marvel, I dug out my father's erstwhile Audels Carpenters and Builders Guide (printed in 1945) to run into if it described the method and if it did, was there an explanation offered. It did, and he did. Read on...
Brand a circumvolve with a square
"Here's a tip for laying out small circles or parts of circles. Tack two nails to set the diameter yous desire, then rotate a framing square against the nails while yous concur a pencil in the corner of the square. Y'all might need to rub a little wax or some other lubricant on the lesser of the square then information technology slides easily. Don't ask us 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 understand the caption.
The Pythagorean theorem is the key, of course, for explaining the reason. For any right triangle:
a2 + b2 = c2,
where 'a' and 'b' are the lengths of the 2 perpendicular sides, and 'c' is the length of the hypotenuse. The same equation also happens to exist (not by coincidence) the equation for a circle of radius 'c,' with the center at point (0,0). So, it stands to reason that if all of the parameters are met (three intersecting straight sides with a correct angle betwixt ii of them), and so the locus of points of all permissible value pairs (a,b) will result in a circle. Information technology 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 ever be the distance between each nail and the 90° vertex. QED
Out of curiosity, I dug out my father'south onetime Audels Carpenters and Builders Guide (printed in 1945) to see if it described the method and if it did, was there an explanation offered. The author did bear witness how to draw a circle with a framing square, and even described how to find the bore of a circumvolve whose area is equal to the sum of the areas of 2 given circles (not certain why that would be need by a carpenter). However, an explicit reason for why it all works out is never given.
Here is what is included in the transmission:
Outer heel method:
Drive brads at points L, F, extremities of the given diameter. With pencil held at the outer heel M, slide square around with its sides in contact with 50, and F, then with the pencil held at M, describe a semi-circle.
Inner heel method:
Evidently if the pencil exist held at S, information technology 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 foursquare sliding on the tacks-the same edges (in either case) that guide the pencil.
At the ends of the diameter LF (fig. 939) drive brads. Place the outer edges of the square against the nails and hold a lead pencil at the outer heel M, any semi-circle can be described as indicated.
This is the outer heel method. simply a better guide for the pencil is obtained by the inner heel method likewise shown in the figure.
FIG. 940 - Trouble ii: To observe the centre of a circle. At the points MS, and LF, where the sides of the square cutting the circumvolve when placed in whatsoever position with heel in circumference, describe diameter and so intersection will exist the eye of the circle. Why?
FIG. 941 - Trouble 3: To describe a circle through 3 points not in a straight line. Permit 50, M, and F, be the given points. Join these points with lines LM, and MF, bisecting them at 1 and 2. Utilise square with heel at one and 2 every bit shown and the intersection of perpendiculars thus obtained at S, will be the center of circumvolve which, with radius LS, may exist described through LM and F.
To find the center of a circle.
Lay the square on the circle and then that its outer heel lies in the circumference. Mark the intersections of the body and natural language with the circumference. A line connecting these two points is a diameter and by drawing another diameter (obtained in the same mode) the intersection of the two diameters is the center of the circle every bit shown in fig. 940.
To describe a circle through three points non in a straight line.
Articulation points with straight lines; bisect these lines and at the points of bisection erect perpendiculars with the foursquare. The intersection of these perpendiculars is the center from which a circumvolve may be described through the three points as in fig. 941.
To find the bore of a circumvolve whose surface area is equal to the sum of the areas of two given circles.
Permit O, and H, be the given circles (fatigued with diameters LR, and RF at right angles). Suppose bore of O, be three inches, and diameter of H, four inches. Then points L, F, at these distances from the heel of the foursquare volition be 5 inches apart equally conveniently measured with a 2-foot rule as shown. This distance LF, or 5 inches, is diameter of the required circle. Proof: LF2 = LR2 + RFii, that is v2 = iii2 + iv2 or 25 = 9+16. (this is as close every bit they come up to explaining the miracle, but non really).
Posted September 1, 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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