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Making Labels for Cones

Date: 06/26/97 at 16:37:17
From: Michael Krone
Subject: To make a label to fit a cone segment

Hi Dr. Math,

I am trying to make labels for my daughter to stick onto different 
sized yogart containers. Because of the shape and angle of the side, 
however, the label never seems to fit properly - it always puckers at 
different points. I want to be able to draw and cut a label that will 
fit perfectly on different sized containers.

I know the diameter, radius, and circumference of the top and bottom 
of the container. I know the distance between the top and the bottom 
of the container. From this, I was able to figure out the slope of the 
sides. But that's as far as I can get!

How do I figure out where the intersecting edges would meet so I can 
take my compass and draw arcs for the top and bottoms?

Or am I going about this in the wrong way? I just want to be able to 
draw the right shape to fit around the container, all the way around, 
if possible.

Thank you in advance for your help with this problem.

Best regards,
Michael Krone

Date: 06/27/97 at 16:18:36
From: Doctor Rob
Subject: Re: To make a label to fit a cone segment

Good question, Michael.  The shape of the unrolled label is as 

The top edge is an arc of a circle. The bottom edge is also an arc of
a larger circle. The two circles are concentric (have the same 
center). The sides of the cone are straight lines which pass through
that common center of the circles. Now all we have to do is compute 
the radii of the small and large circle, and tell you the angle 
between the two straight lines.

Let the radius of the small circle be r, and that of the large circle
R. R is the slant height of the entire cone, and r is the slant height 
of the part of the cone that is missing. The distance from the vertex 
of the cone (extended) to the plane of the circle at the top of your 
cone will be h. Then the distance from the vertex to the bottom is 
h + H, where H is the perpendicular distance from the top to the 
bottom. Let s be the radius of the top, and t the radius of the 
bottom. Then there the following equations are derived from similar
triangles and the Pythagorean Theorem:

  h/(h + H) = s/t = r/R
  R^2 = (h + H)^2 + t^2
  r^2 = h^2 + s^2

You already know H, s, and t.  From the first equation, we get:

  h = s*H/(t - s)

Using this, you can compute both:

  R = Sqrt[(h + H)^2 + t^2]
  r = Sqrt[h^2 + s^2]

The angle in radians is the arc length divided by the radius:

  A = 2*Pi*s/r radians

To convert to degrees, multiply by 180/Pi:

  A = 360*s/r degrees

Using the same center, draw the two circle arcs with radius r and R. 
From the center draw two radii making angle A. The area between the 
circles and between the two radii should fit the surface of the 
frustum of a cone which is the shape of the yogurt container.

If you need more information, write back to us again.

-Doctor Rob,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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