GC-10 Flattening The Circumference Of A Circle

Here's how you can flatten a circle's circumference:

Needed analogue tools:

• Dividers

• Pencil

• Straight Edge

• Eraser

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Given:

• Circle A-B, Ø=D=10

Desired:

• Circle A-B's circumference, Line A'-D

Note: Refer to GC-09 on how to divide a line in N-segments.

1. Draw Line A-B which represents Circle A-B's diameter.

2. Draw a straight line starting from A and divide it in 6 equal segments. Refer to GC-09.

3. Draw another straight line starting from B but in a direction opposite to, and parallel to the line in Step 2.

4. Using the dividers, split the line in Step 2 in 6 equal sections by marking the points c, d, e, f, g, h. A-c=c-d=d-e=e-f=f-g=g-h. The spacing you choose is up to you or as is most practical, given the paper space you have at hand.

5. In a similar fashion, divide Step 3-line in 6 equal sections, whose lengths are the same as the ones we created in Step 4, i.e. A-c=B-i, i-j=j-k, k-l=l-m, m-n.

6. Using a straight edge, connect the dots in the following manner:

   1. c-n

   2. d-m

   3. e-l

   4. f-k

   5. g-j

   6. h-i

7. The line segments just created (Step 6) intersect Line A-B at 6 locations:

   1. o

   2. p

   3. q

   4. r

   5. s

   6. t

8. Thus, distance D=A-B=A-o + o-p + p-q + q-r + r-s + s-t + t-B; A-o=o-p=p-q=q-r=r-s=s-t=t-B=1/7=0.1428 . Moreover, A-B=D=7*A-o

9. Place one leg of the dividers on A, the other on o. Maintain that stretch.

10. While one leg sits on A, swing the other (while maintaining A-o stretch) to the left of A, opposite of o.

11. Mark that new point as A'.

12. In a similar fashion measure off A-B, circle's diameter. Maintain the divider's spread.

13. Place one leg on B, the other to the right of B.

14. Mark that new location as C.

15. Mark off one more point, divider's spread unchanged, to the left of C.

16. Mark that point as D.

17. Thus, the new line/distance A'-D approximates very closely the true circumference's length.

The graphical solution gives a distance of A'-D=31.4285, the calculated one gives 31.4159. That's an error of 0.05%. So, it is safe to assume that such a graphical construction is accurate enough to be applied in most carpentry tasks.

The earliest records of such approximation was drawn in Mathes Roriczer's "Geometria Deuetsch" manuscript published ca. 1497 A.D.

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You can download the CAD and GH file here:

10_Flattening_The_Circumference_Of_A_Circle

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10_Flattening_The_Circumference_Of_A_Circle

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