Deci-Lon and Areas
I was recalling a phrase in an earlier post, where the phrase, “When the area is known, reverse for diameter” was used. I was thinking, perhaps, to expand on that somewhat vague statement. What decided me was thumbing through a rather older mathematics book called ‘Practical Mathematics’ (Palmer 1919) written the last century, where mathematics was actually taught to students, and they were expected to understand what it all meant.
In any case, I was flipping through it, and saw some interesting formulas for area. Understand, this is easily discernible with algebra and re-arranging a formula, but this was the first time I had actually seen it in print. This particular formula would be handy in a situation where someone needed to construct a gazebo or tank, or any circular structure to encompass or enclose a certain area. Then I noticed it was easily performed on the slide rule, in this case the K&E Deci-Lon slide rule.
The formula is not unknown, but it is more obvious when noted down as here,
\[D = \sqrt{A \div \frac{\pi}{4}} \quad \Leftrightarrow \quad \sqrt{A \div 0.7854}\]
where D is diameter, A is area, and 0.7854 is a small tick mark on the A, B, C and D reverse-side scales of the Deci-Lon, and many other slide rules as well.
The particular book I am using is a reprint of the above referenced book, printed in 1988 by Lindsay Publications. The formula is shown on page 152 of that book.
Now, just for fun, let’s do a couple of examples. We want a backyard fire pit inside a circular area of 500 ft2. So, we set the hairline (HL) to 500 on the D scale, slide the C scale 0.7854 tick mark under the HL. We then move the HL to the index, flip the rule to see the Sq scales. As we have three digits, we read the upper Sq1 scale for diameter of 25.2 ft. We use the upper Sq scale for odd numbers of digits, and the lower Sq scale for even numbers of digits.
Another useful rendition is,
\[r = \sqrt{A \div \pi}\]
where r is the radius. This will allow us to easily inscribe a circle of desired area simply by drawing a circle from the center. Suppose we needed a 12 ft.2 fire pit in the center of the above area. Presume we have already marked the center, we just need the radius that would enclose 12 ft.2 So we set the HL to 12, slide 0.7854 tick mark under the HL, move the HL to the index, and flip to read the Sq2 scale for radius of 3.91 ft.
The same procedure also works on many other slide rules. For example, the Post Versalog 1460 has the square root scales, called R1 and R2, on the stock below the D scale on the “front.” There are no tick marks on the Post however, so there is no reason to flip to another side. Of course this also applies to the K&E, where flipping is only required if we desire to use the tick marks.
So, that’s enough for this time. Take care, God Bless!