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Dividing a Circle into 8 Equal Parts

Date: 08/19/2002 at 22:23:17
From: Thomas Weir
Subject: Dividing a circle into equal parts

If you have a diameter (= 14ft), can you mark off the 1/8 sides of the 
circumference so that they are equal?

Thank you,
Tom

P.S. I'm a contractor, and the concrete is coming at 1:00 P.M. 
tomorrow.


Date: 08/20/2002 at 09:15:49
From: Doctor Rick
Subject: Re: Dividing a circle into equal parts

Hi, Tom. We're glad to be of help to people who need math in their 
real-world jobs! Many students ask us, "When will I ever use this 
stuff?"

Are you saying that the circle with diameter 14 feet is marked out, 
and all you need to do now is divide the circumference into 8 equal 
segments?

The most direct method would be to use a tape measure to lay out a 
length of 1/8 the circumference along the arc. Since the circumference 
of a circle is pi (about 3.1416) times the diameter), this distance is

  1/8 * 3.1416 * 14 feet = 5.498 feet = 5 feet, 5.97 inches

Measuring along a curved line isn't easy, so we can do better. We can 
find the *straight-line* distance between the marks on the circle. 
Imagine that we have the marks and we connect opposite marks with 
diameters of the circle. There will be 4 lines, and each will make an 
angle of 45 degrees with its neighbors. Now connect the neighboring 
marks on the circumference; we have made 8 isosceles triangles (two 
sides the same length, equal to the radius of the circle). We want to 
find the length of the third side of this triangle, knowing the other 
sides and the angle.

I won't go into the trigonometry involved in the next step unless 
you'd like to see it. The answer is:

  L = D*sin(45/2 degrees)

where L is the distance we seek and D is the diameter. Using a 
calculator to find the sine of 45/2 = 22.5 degrees, I get:

  L = D*0.382683

With a diameter of 14 feet, L = 5.3576 feet, or 5 feet, 4.29 inches. 
Notice that this is a bit less than the distance around the curve: a 
straight line is the shortest distance between two points.

The upshot is this: Make the first mark on the circle, then measure a 
straight-line distance of 5 feet, 4.29 inches from there to find the 
next point on the circle. You could keep going this way all the way 
around (probably ending up with a little error at the end); or if you 
know where the center of the circle is, you might make just 4 marks, 
then extend diameters from these 4 points through the center of the 
circle to find the other four points. You probably know better than I 
how to minimize the errors in measurement. (My house projects don't 
come out that well!)

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Euclidean/Plane Geometry

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