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 Question from Stephanie, a student: I'm trying to make a cone out of a flat sheet of metal for a "claw setting" for a gemstone. The cone must be 8mm wide at the top and 11mm long tapering to a point. But because the prongs must be cut out of the top the cone should not start to taper for a length of 3mm from that top 8mm. The 3mm prong is then bent over the 8mm stone. That probably doesn't make enough sense. But I don't know how to explain it. If it helps a claw setting is the very common prong setting for engagement rings or earrings. Please help as soon as possible as this is a commissioned piece for someone and I'm running out of time. I don't remember any math really from high school so please make the instructions really easy to follow. Thank You!!

Hi Stephanie,

First consider the cone without the claws so the cone is 8 mm long. We will worry about the claws later.

I am not sure what measurement you are calling the length. Is it h or s? h is called the height and s the slant height.

I am going to assume that the h= 8 mm.

If you slice up the side of the cone and roll it out flat you have a sector of a circle of radius s mm.

Triangle ABC is a right triangle and Pythagoras' theorem then tell us that

|AB|2 = |BC|2 + |CA|2

thus

s2 = 42 + 82 or s = 4 √5 mm

Thus the distance |AP| = 4 √5 mm and the arc from P to Q is the circumference of the circle at the top of the cone which is 2 r = 8 mm.

We need the angle at A which is a fraction of 360 degrees. The length of the arc PQ is also a fraction of the circumference of the entire circle of radius 4 √5 mm. This circumference 2 r = 8 √5 mm. From the symmetry these fractions are the same which means

(angle at A)/360 = 8/(8 √5)

thus

the angle at A measures 360/√5 = 161 degrees.

Draw a circle of radius 4 √5 = 8.9 mm and mark off a sector with the central angle 161 degrees. Draw another circle of radius 8.9 + 3 = 11.9 mm with the same centre.

Cut the claws from the band between the two circles and roll it up to form the cone and the claws will be 3 mm long.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.