In article ,
John Moorhead wrote:
Folks -

*Somewhere* there was an online calc or website that was able to figure out
the length of the segments in creating a circle of "X" inches in diameter.
I am sure that many of you math types will guffaw, so I'll take my lumps,
but I'd like a better way than laying it out on paper and measuring.

The problem is not exactly trivial, but almost.

The length you want is simply "double the sine of the half-angle".
("sine" takes a trig table, or a calculator.)

Here's _all_ the gory details of what and how:

Obviously the paper-and-pencil method is to draw a circle, draw the required
number of radius lines, and "measure" the straight-line distance between the
end-points of two adjacent radii.

Now, consider those two adjacent radii, and the portion of the circle that
connects them.

Draw the chord (the straight line that you want) that connects the ends
of the radii.

Draw a _third_ radius line that goes down the middle of the piece.

Note that it will _always_ intersect the chord at a 90 degree angle.

And that the angle between that new line, and either of the 'edge' lines
is exactly half the angle of the whole.

These relationships are true *regardless* of how many sides the polygon
you're constructing will have.

Now, considering one of those _right_triangles_, you know the length of
the hypotenuse, *and* the angle at the circle center.

This is everything you need to know to _calculate_ dimensions.

the length of the chord is, obviously, twice the length of the side of
the right triangle.

the length of the side of the right triangle is computed as the 'sine of
the angle opposite it, multiplied by the length of the hypotenuse'.

This does require a 'trig table', or trig functions on a calculator.

Now we've already established that the 'opposite' angle of the right-triangle
is 1/2 the angle of the full chord. so we just calculate

(sin(1/2 angle)*hypotenuse)*2

commonly described as "double the sine of the half-angle".


e.g. for an 8-sided 12" diameter circle:

a single side subtends 45 degrees
so half that angle is 22.5 degrees
for 12" diameter, the radius is 6"

Thus:
sin(22.5 degrees)*6"*2
(my handy-dandy calculator says that "sin(22.5 degrees)" = 0.38268)
= 4.592 inches (or approximately 4 19/32")


The MS-Windows "calculator" (Start-Programs-Accessories-Calculator) does
trig functions, if you click on the 'View' button and select "Scientific'.

"Conveniently", it takes angle input in 'degrees', by default. you don't
have to go through the nonsense of converting to 'radians'.





To 'confuse' people, you can just multiply the 'half angle sine' by the
diameter. grin