Making Cone Frustums (Revised)

Today's lesson is on how to make approximations for spherical objects out of paper. Why? It's easier and lighter to make these to fill up the dimensions for a round part of the model (in my case, to round out the joints to make them more spherical) than by making them by layer.

Figure 1: Making something round into a polygonal mess! The item on the far right is the side view of this "frustum" I'll be talking about.

I've taken the essence of the idea from a paper modeler forum.

(Edit: URL doesn't exist anymore. No point making you click on a link that doesn't work. I might as well explain how it works here.
-J.Norad)

The principle is that you draw two concentric circles, cut out a sector, and when you glue that sector end to end, you get a cone frustum. How does it work?

Figure 2: All you need to know about the next paragraph.

First, you need to know three things: how tall your cone frustum is, and the smaller and larger diameters that compose the top and bottom of it. From Figure 2, you can see the arc segment for the smaller and larger circle shares the same arc angle. As a refresher, arc segments are:

Eq 1.) arc segment length = pi * radius* 2 * (arc angle)/(360 degrees)

We're going to use this to set up two equations: one being the arc segment length of the smaller circle and the arc segment length of the larger circle, then setting the angles equal to each other. This gets us

Eq 2.) (Arc length 1)/ (π*radius1* 2) = (Arc length 2)/(π* radius2 *2)

or simplified,

Eq 3.) (Arc length 1)/ (radius1) = (Arc length 2)/(radius2)

We know arc lengths 1 and 2, since we've chosen what the cone frustum's top and bottom diameters are. These diameters will correspond to the arc lengths, since when we cut and assemble the sector, the edges will form the diameters of our frustum. Confused? You'll understand when you try it out soon.

As a recap, the circumference of a circle = pi * diameter.

However, before we can proceed, we need to know the difference between the two radii of the concentric circles we're going to draw. This is where the height of the frustum comes into play. We'll be using the Pythagorean theorem to do this.

As a refresher, the hypotenuse of a right triangle is equal to the square root of the sum of the squares of the two legs. (a^2+ b^2 = c^2)

Figure 3: Frustum geometry and how it relates to our arc sector.

We want the length of the sloped side of the cone frustum. We have the height and the bottom leg, which is HALF the difference between the two diameters since we have two sides of the trapezoid. Substituting these into the Pythagorean Theorem and referencing Figure 3, we get the following:

Eq 4.) Radius2-Radius1 = square root of (height^2 + ((larger diameter- smaller diameter)/2) ^2)

Hooray. Now we combine Equation 4 with Equation 3 and we can solve for one of the radii. Once we know what one radius is, we can determine the other using Equation 3, since we know the arc lengths= circumference of our cone frustum tops and bottoms. Once we know the radius and arc length of the inner or outer circle, we can solve for the arc angle using Equation 1.

Still confused? Can't be bothered solving equations? I have simplified the process down for you, with less potential error involved by turning it into a Microsoft Excel spreadsheet!

Edit: Added a download link to a much more updated cone calculator. This one's a bit more silly, as it includes my attempt to calculate the shape of offset cones.

Figure 4: Excel spreadsheet. Copy as directed.

What you'll need to do to use this is enter the smaller frustum diameter and the larger frustum diameter along with how tall you want the section to be. It will spit out two circle diameters which you should then draw concentrically, along with an angle. Use the angle to draw a sector on those circles. The resulting rainbow shaped arc will be the thing you need to cut out and glue together. Just remember to add your own tabs.

Here's the layout of the spreadsheet. Fill in the text fields as they are arranged, as the commands reference the numbers in the second column exactly.

Now paste in
B5: =B2/2*SQRT(1+4*B3^2/(B2-B1)^2)
B6: =B5-SQRT(((B2-B1)/2)^2 + B3^2)
B7: =B2*360/(2*B5)

E5: =B5*2
E6: =B6*2
E7 was a reference number for me, since some of the angles produced were over 180 degrees and I only needed to deal with the remainder. This value is =b7-180 in case you want to reference them in terms of increments of 180, since protractors tend to deal only with 180 increments.

Figure 5: Cone frustum side view on left; sector ready to cut and glue on the right

This image best shows the result of the exercise. Stolen from the site I linked from.

Once you try it out, you'll understand what's all behind this gibberish and can start making cones out of paper.

(-Revised by J.Norad, June 16/2009)