PROBLEM:
Jim,
How do I make a pattern for a cone that is 12 inches tall and has a base diameter of
24 inches and a top diameter of 8 inches. Can you share the formula with me
please? I know how it looks but it would take a lot of trial and error to get the
measurements right. It is a transition for some pipe flues.
Thanks, Carl Condray, West Texas
If Carl wanted a full cone rather than a truncated one, we could easily develop a net
for the cone like so:
where a is the
slant height of
the cone and
a
the angle t is to
be determined.
a
t
h
d1
We want the black arc on the net to have length equal to the circumference of the
cone’s base. Thus:
( )(2π a) = π d
t
2π
1
∴ t=
πd
1
a
(in radians)
But Carl wants a truncated cone. All we really need to do is make the full cone, then
cut a piece out of the net’s center (in the shape of another cone) to make Carl’s cone.
Let’s make a slice through the cone to give us two right triangles, as shown:
(Let’s work in the general case, with
larger diameter d1, smaller diameter
d2, and height h. Then we can plug
d2
in Carl’s specific values later.)
h
d1
Let’s label some pieces of the
triangles’ extensions, and then figure
out what we need to know.
Since the bases are parallel, we know that
ΔABC ~ ΔDEC. Thus,
C
a- y
x
d2
x+h
E
x
=
x+h
=
x
d  d 
d
d
2 2
   
∴ ( x + h)(d 2 ) = ( x)(d1 )
1
D
y
h
∴
2
1
2
∴ x(d1 − d 2 ) = h ⋅ d 2
d1
B
A
∴ x=
h ⋅ d2
d1 − d 2
Now let’s determine the outer ((a – y) + y = a) and inner radii (a – y) of our net.
These are the slant heights of the overall cone and of the removed cone, respectively.
By the Pythagorean Theorem, we have:
1
a = ( x + h) 2 + 2 d1
and
1
a − y = x2 + 2 d2
We know d1, d2, and h, so we can calculate y and b = a – y easily.
What is the angle t between the two sides of the net? Since this depends only on the
circumference of the cone’s larger base, we can compute it just as we did for the
nontruncated cone:
( )(2π a) = π d
t
2π
1
∴ t=
πd
1
a
(in radians)
These formulas are used in the GSP file included with this solution to give you the
important dimensions of a net, given d1, d2, and h. Specifically, for Carl’s data:
d2
t
h
a
d1
d 1 = 24.00
d 2 = 8.00
h = 12.00
d1 ≥ d2
a = 21.63
b = 7.21
t = 199.69°
b
Here is a cone that I made for Carl (scaled down). I printed the net from my GSP file,
cut it out, and taped the edges together.
With the GSP file, I can make these patterns just by entering the two diameters and the
height, printing, and scaling by the percentage given by the file.