How to Calculate the Template of a Cone's Portion

I often make cardboard accessories and I often need some conic parts. Recently I made lightsabers and, once more, I took ages trying to remember how to find the proper angle and radius for the template. Yes, it may seem easy to logical people but sometimes the brain just doesn't want to put the effort into it so... here is an explanation and a little program to do it easily on your own or to let the program do it for you and save time ☺.

To calculate you will probably need a calculator. Then to trace you will need a compass, a ruler and a protractor. Finally, a little bit of paper (and glue, or tape) to build your cone's portion.

To explain easily, we will have to use letters to talk about the numbers we are using. Then, we can go through the steps, calculating one thing after the other.

a- the little radius that you want on your final piece

b- the big radius that you want on your final piece

c- the high that you want for your final portion of cone

d- the difference between a and b → b-a

e- the height that the cone would have if it was complete

Now we'll use Thales' theorem in order to find the height that the cone would have if it was complete, using the relation: d/b = c/e. To find e we do a cross multiplication: e= (c*b)/d 

f- the big radius that you will trace

This time we'll use Pythagoras' theorem in order to find the hypotenuse of the triangle. This correspond to the radius we will have to trace. Pythagoras' relation is the following : f² = e²+b² so f = √ (e²+b²).

g- the hypotenuse of the little triangle

Now we'll do the same but with the little values, to find the hypotenuse of the little triangle : g = √ (d²+c²)

h- the little radius that you will trace, aka the difference between f and g → f-g

i- the angle of circle we will need : we are searching for the proportion of the whole circle we need to have the final circle. The proportion is b/f (or a/h it should be the same). Let's just do (b/f)*360, to find the proportion we need in degrees.

To conclude, you need to trace two circles (with the same center) with a radius of f and h, and then draw the angle of i degrees!

Now we have everything we need to trace the template of the cone's portion, cut it and assemble it!

To make things clearer, let's have a look at an example.

a- I want the radius of the little circle of my final cone's portion to be 4 cm.

b- I want the radius of the big circle to be 6 cm.

c- The portion will be 3 cm high.

d- The difference is b-a = 6-4 = 2 cm

e- The height of the complete cone would be e= (c*b)/d = (3*6)/2 = 9 cm

f- The big hypothenuse is f = √ (e²+b²) = √ (9²+6²) = √(117) ≃ 10,8 cm

g- The little hypothenuse is g = √(d²+c²) = √(2²+3²) = √(13) ≃ 3,6 cm

h- The little radius is h = f-g = 10,8-3,6 = 7,2 cm

i- The angle is i= (b/f) * 360 = (6/10,8) * 360 = 200°

In this exemple, I need to trace two circles (on the same center) with radiuses of f=10,8cm and h=7,2cm. We will only take 200° of the ring, glue it, and have the dimensions we wanted.

Now if you're not motivated to do several calculations you can simply use this program. It's written in Python (a common programming language), so you need a Python website to read it.

Here is the link to a website that allows you to read Python programs: https://replit.com/languages/python3

And here is the program you have to copy-paste in the left part of the website:

from math import *

def cone(a,b,c):

 d=b-a

 e=(c*b)/d

 f=sqrt(e**2+b**2)

 g=sqrt(d**2+c**2)

 h=f-g

 i=(b/f)*360

 return "the radiuses are",f,"and",h,"the angle is",i

Then press ►Run

For the console to tell you the numbers, you need to ask for them in the right part of the website, by simply writing: cone(a,b,c) and replacing a,b,c by your values. A is the little radius that you want on your final piece, B is the big radius, and C is the height of your cone's portion. In the example of step 2, i would've written: cone(4,6,3) and then press enter. It gives you three numbers, the last one is the angle.

We're done ! I hope you managed to understand everything beside my (sometimes) poor English, please tell me if anything seems unclear.