## Math pop quiz

Just when you think you only need high-school math to figure out the losses on your 401(k), along comes a reader with a question. Barry Randell plans to build a cover for a 16″-diameter cistern. The cover would have six or eight sides, and he wanted to know how long the pieces would be. My plan was to draw an octagon on paper, so here’s how that worked:

octagon16

1. You know a 16″-diameter circle (shown in blue on the illustration) can be captured in a square with 16″-long sides, so draw a square that size using a yardstick and framing square.

2. Draw diagonal lines between opposite corners of the square to find the center at the intersection of the lines.

3. Set a compass for the distance between one corner and the center. (You can make a crude beam compass by clamping a scratch awl and a pencil to the yardstick 16″ apart.

4. Place the point of the compass at one corner and mark where the arc (shown in red) intersects two sides of the square. Do this again at each of the remaining corners until each side has two arc marks on it.

5. If you want to stop there, you can measure the distance between two arc marks on any line to discover the length of the inside face of a mitered frame piece. Or connect the arc marks closest to a corner and repeat at each corner to create the octagonal shape.

Now, in fairness, Barry did ask for a formula and not a geometry exercise. So I’ll open the question up to anyone whose math skills are better than mine. In other words, everybody. And in the words of my high-school math teachers, remember to show your work.

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11 Responses to “Math pop quiz”

This was high school geometry when I was a kid.

Call the length of the regular polygon (octogon in this case) side “s”. Regular means having equal length sides, not, um…

Draw a line from the circle center to one of the corners of the regular polygon. Call the length of this line “R” (it’s the radius of the circle).

Draw another line from the circle center to the exact middle of the side whose length you wish to know.

If you’ve drawn this correctly, you now have a right triangle with the hypotenuse (longest leg) R, shortest leg of 1/2s, and smallest angle “a”.

Angle “a” is 180 degrees divided by the number of polygon sides (8 in this case, or 22.5 degrees).

Trigonometry tells us that:

sin a = opposite/hypotenuse or sin a = (1/2s)/R

SO if we solve this equation for s we get:

s = (2R)sin a or, in this case:

s = (16)sin 22.5 = 6.123″ or VERY close to 6 1/8″

This method works for all REGULAR polygons.

2. There is another way to do this with out a compass.
1. Draw a 16″ square

2. Connect the corners diagonally with a line

3. Measure to the center of each side of the square (8″) and put a mark

4. Meaure 8″ from the center of the square on the diagonal lines and put a mark

5. Connect the marks from the center of each side of the square with the marks on your diagonal line and you wind up with eight equal sides

3. The method given by Joe Thompson will not work. It is true that the center point of each side of the square is the circle. It is also true that the point on the diagonal (8 inches from the center) is on the circle. But connecting those points will leave you with an “inscribed polygon” (it is inside the circle) and there are gaps around the polygon as the arc of the circle goes above this line.

4. There is a lot to be said for “ruler and compass geometry.” There are lots of techniques that are dieing with the carpenters and cabinetmakers of the pre-power tool days.

When I was in graduate school (mathematics), in one “abstract algebra” class, we had spent several weeks on a highly technical and somewhat obscure class of sets and their properties. The professor concluded the last class by saying, “And, that’s why you cannot dissect an angle with ruler and compass.” All the students sat there looking at each other with puzzled looks. This trisection was one of those problems that attracted crackpot solutions for a century, with some even being published as paid ads in newspapers. The next class explained why this theory of abstract objects applied to the trisection problem. Oh, now I get it.

6. A few years ago I was building my kitchen table and had the same question about how to calculate the length of the inividual boards, hoping that what I already had in my wood stock would suffice. I asked my machinist brother how to do it, and he said there were two solutions: one involved calculus, which I never studied. The other involved a piece of chalk, my garage floor and a tape measure. Draw the circle, measure the length of each chord across the circle. I opted for the chalk method…..

7. Bob Pettit is agonizingly close, but will be disappointed when he constructs his octagon. The diameter of the cistern is given as sixteen inches, which is the distance across the flats of the octagon, not across its points. Consequently, if the side of the octagon is ‘s’, then the tangent of 22.5 degrees is 1/2s divided by 8 inches, and the length of each side becomes 6.627 inches, or very nearly 6 5/8 inches. Bob’s lid will balance on its points, leaving an arc of the cistern exposed between each point of the octagon. It might be good enough, but if he elects to go with a hexagon, he will likely have gaps between the edges of his lid and the cistern walls. The length of the sides of the hexagon should be 9 1/4 inches, while Bob’s method will make them only 8 inches.

I used to have a copy of a book called “The Steel Square” which offered all sorts of exotic solutions to common problems. Ever since calculators came out, though, I haven’t needed the book, and I don’t know exactly what’s happened to it.

8. For those who are familiar with a center marking gauge that slides 2 dowels along the sides of the piece and a pencil marking the center, boatbuilders for years have used a similar device for marking square stock to reduce it to 8 sides. The distance between the 2 dowels and the 2 pencils (or scribes) is simply 7-10-7, whatever units you want to use. For something as simple as a cistern cover, this will get you closer than you can cut it with a circular saw.

(The simple math says that 7 squared times 2 equals 98, and the square root of 98 is “almost” 10.)

9. There is an old school way to do this that is actually the simplest and most accurate.
First, the lid should cover the hole by about 2 inches on each side; this makes the lid 20 inches across at the shortest point. Lay out a 20 inch square. Mark the center of each side and draw crossed center lines (not diagonals).
Now, dig deep into your collection of old tools and find your framing square. If it is a quality square, in the middle of the face on the tongue you will find a row of dots down the middle. This is the octagon scale. The dots are a scale representing the length of the side of a square in inches.
With a pair of wing dividers, put one point on the first dot on the octagon scale (closest to the heel) and the other point on the twentieth dot. The dividers are now set to the distance from the center of each side to the corner points of the octagon (half the length of each side of the octagon). Using the dividers, mark these eight points, two along each side of the 20 inch square. Now connect these points across each corner of the 20 inch square and you have a perfect octagon that fits.
To finish the job, attach a 16 inch diameter circle or some cleats to the bottom of the lid to prevent sliding.

10. I love all the descriptions about how to draw an ocotogon. My favorite is using the framing square. Real old school and cool too.