shape of tapered bottle.

I wonder if anyone can help me. I need to design a sticker to cover a jar which will be used to collect money for charity. The relevant part of the jar is 405mm in circumference at the top, and 390mm in circumference at a point 85mm lower. How can I plot the shape to use as my outline on my computer, so that when printed and cut-out and stuck on it will be straight all round at the top of the jar?
Hope I've made it all clear.

Hello, bigginge!

Quote:

$\text{Design a label to cover a jar used to collect money for charity.}$
$\text{The jar is 405 mm in circumference at the top, and 390 mm}$
$\text{at the bottom at a point 85 mm lower.}$

$\text{How can I plot the shape to use as my outline on my computer,}$
$\text{so that when printed and cut out and stuck on,}$
$\text{it will be straight all around at the top and bottom of the jar?}$

We want a sector of an annulus (ring).

Code:

* * *.. * |:::::* 405 * |:::::::* * *:: 390 :* * | *::::: * * |@ *::::* * * *----*----* * *- r * 85 * * * * * * * * * * * * *

The radius of the smaller circle is $\,r.$
The sector has central angle $\,\theta.$
The arc length on the smaller circle is 390 mm.

The radius of the larger circle is $\,r+85.$
The arc length on the lartger circle is 405 mm.

Arc length formula: . $s \:=\:r\theta$ . with $\,\theta$ in radians.

Smaller circle: . $r\theta \:=\:390$ .[1]

Larger circle: . $(r+85)\theta \:=\:405 \quad\Rightarrow\quad r\theta + 85\theta \:=\:405$ .[2]

Substitute [1] into [2]: . $390 + 85\theta \:=\:405$

And we have: . $\theta \:=\:\frac{3}{17}\text{ radians}$

Substitute into [1]: . $r(\frac{3}{17}) \:=\:390 \quad\Rightarrow\quad r \,=\, 2210$

Construct two concentric circles with radii 2210 mm and 2295 mm.
Cut out a sector with central angle of $\frac{3}{17}$ radians or about 10.1 degrees.

The label is the curved region between the two circles.

Thank you! It's been about 45 years since I got my maths O level and I didn't understand the first part, but I did the summary in Adobe Illustrator and it looks just right.
Thanks again.