View Full Version : Determining the Angle of a Sector

04-06-2007, 07:11 AM
I had a problem and I think my solution might be of use to others. I had to determine the angular section of a piece of pipe. Without the center of the pipe, to measure from, it looked like a difficult problem. The situation is shown in Fig. 1.

To solve the problem, I went back to geometry. I traced the outline of the pipe on a piece of paper and marked the ends of the angular sector on the traced circle. I drew a line between the two marked sector ends. I measured the line and marked the center point. See Fig. 2.

From this measured center point, I used a square and drew another line perpendicular to the line until it intersected the edge of the drawn circle. See Fig. 3.

From this intersection with the circle, two lines are drawn back to the ends of the angular sector. See Fig. 4.

The angle between the two lines just drawn is measured. The measured angle is exactly one half of the angle of the sector made with the center.


Forrest Addy
04-06-2007, 09:40 AM
Sorry to spoil your discovery. Your technique was faultless except sector angles are referenced from the center of the radius not the diameter opposite the chord center.

Also while geometric construction technique works very well for determining unknowns for almost any shop layout problem the mathematical methods are more accurate - if accurate data is available to start with. Yours is a problem where the chord of a sector and chordal height is available and possibly the stretchout length of the sector. Using the formulae it's possible to determine the radius and the sector angle to an accuracy consonant with the starting data.

Here's a cool calculator for XP users:


04-06-2007, 10:03 AM
If you have a handy scientific calculator then just measure distance A.

Calculate the ratio B= (A-radius)/radius

Find the arcsine (inverse sine) of B and you have the sector angle.


04-06-2007, 10:03 AM
Here's my take on how to do this:

(I've corrected the image to reflect Mklotz's comment that I missed a factor of two when writing this up.)

04-06-2007, 10:03 AM
Forrest, Yeah but he's talking about pipe and his method could be useful there. Of course, one could also use a PI (or actually any) tape for rough work (and only 4 function calc available) and a dividing head for any precision.

Tape approach:

angle = tape reading (sector length) X 360 / tape reading (entire circumference)

or, rearranging if you need the sector length for cutting or marking:

tape reading (sector length) = angle X tape reading (circumference) / 360


04-06-2007, 10:10 AM
As Forrest said, the ANGLE of the sector is figured from the center of the diameter, not the opposite side of the diameter.

04-06-2007, 10:32 AM
Bigboy's construction is also correct.

04-06-2007, 10:32 AM
Hmmm....when I plotted the thing in CAD the results agreed with BigBoy1's method. So, what's the problem guys?

04-06-2007, 10:44 AM
No problem, some have a surface plate under the pipe and some have mud :D

04-06-2007, 10:45 AM
Here is the limit case. The chord is perpendicular to the line that intersects the half angle on the opposite circumference.


Upon further consideration, here is the actual limit case.


04-06-2007, 11:21 AM
Evan, what are you using to draw those pictures?

04-06-2007, 11:25 AM
Can we trisect the angle now? Please?



The Doctor
04-06-2007, 11:37 AM
Forrest, nice calculator. Works in Win 2000, BTW:)


Lew Hartswick
04-06-2007, 11:43 AM
Here is the limit case. The chord is perpendicular to the line that intersects the half angle on the opposite circumference.


Upon further consideration, here is the actual limit case.


I think you can even extend that "limmiting case" to a complete 360 deg.

04-06-2007, 12:14 PM
For simple drawings like this I use Paint Shop Pro. It has some CAD capability that most people don't realize exists. I can draw much faster with it than with a pencil.

Forrest Addy
04-06-2007, 12:58 PM
Bigboy mentions "sector angle". I took him at his word and pointed out what I thought to be a flawed concept in his drawing.

Not to get pedantic but this is plain geometry. A sector is defined as a portion of a circle bounded by two radii and the included arc. It's a closed plane figure having area like a piece of pie.

A circle is a line segment rotated in a plane about a point. An arc is a segment of a circle. It has no area because it is not a closed figure.

I don't know why everyone is drawing angles from a diameter bisecting an arc and associating that with "sector." It might be an illustration of micrometer technique but it's not a sector.

The limiting case in Evan's figure continutes in infinite steps as the 90 degree included angle between the two line segments increases and approaches 180 degrees and the included angle of the arc approaches 360 degrees. As this continues, line segments shorten to coincident points on the vertex and the circle finally closes. Conversly you dan dial theevolution backward to where the arc's included angle approaches zero the line approach true diameters and that angle between them approaches sero. There again the limiting case resolves to a point but one representing what was once a one limit of a diameter and a radius.

Euclid is spinning as we post.

04-06-2007, 01:19 PM
Forrest, I guess I was reading the original poster's intent as a means of marking a sector at the surface of the pipe. Many portable means of cutting away a section of pipe would do so perpendicular to a tangent line at the point of contact. Of course, this would project to the center of the pipe as you have indicated.

I suspect Euclid has heard worse ;)


04-06-2007, 01:38 PM
Here's my take on how to do this:


Unless I'm missing something, this gives one half the angle subtended by the chord.

04-06-2007, 03:05 PM
I consider the 90 degree included angle as the limit case because it defines the maximum possible chord, i.e., the diameter.

04-06-2007, 03:46 PM
I was thinking about this at breakfast after I posed my little diagram and you're right. I had diagrammed theta on the whiteboard as half the needed angle for purposes of calculation and then forgot to multiply the angle by 2. I'll change my original post to have the right drawing. Oops.

04-06-2007, 04:59 PM
I'm sorry that I didn't state my problem more clearly. I had to determine the angle subtended by a section that had to be cut into the side of a pipe. Since I didn't have an opening, the technique for determing the angle with an opening does not apply but is a good thing to keep for future reference.

My technique to cut the opening in the pipe was to mount the pipe horizontally in mill, with the one end of the pipe mounted in the rotary table turned vertically and the other end in a tailstock. I used the rotary table to turn the pipe through the required angle while milling. This ensured that the sides of the section removed were "flat" to the sides of the pipe and the required angle was acheived. Tool offset had be be worked into situation so that the desired angle would be acheived. It may not be the "best" to solve the problem but it worked for me. "There is more that one way to skin a cat."


04-06-2007, 08:00 PM
This is the way I would do it. It should work.
Diameter x pi = circumference

Circumference / 360 = amount per degree

Make plunge cut and rotate rotary table the amount desired minus
The cutting tool diameter.

If cutting tool is 5/16 (.312)
Pipe OD is 1 then 1x 3.1416= 3.1416/360=.0087266 per degree

45 degrees x .0087266=.3927 minus tool diameter of .312
Move rotary .0807/ or 9.247 degrees
Cutting tool at 5/16 is equal to 35.752756 degrees

Tool equals 35.752756 degrees , tool movement equals 9.247 degrees combined total = 44.999756 degrees.