Pattern for a Truncated Cone:

One Solution

First, think about extending the conical figure up to a point. That is, this pedestal is formed by cutting off a cone with a base having a diameter of 8 inches. The completed cone looks like the figure at the right.

I have labeled a few lengths in the figure because they will be useful later.

The idea is to know the values of h, a, and b and then roll the figure out flat.

First, h. We have h + 12 is the altitude of the whole cone and h is the altitude of the little cone. Now, we have similar triangles and so

So, h = 6 -- that is the altitude of the little cone -- and the altitude of the big cone is 18.

By using the Pythagorean relation for the right triangles, we can determine the lengths of the lateral sides of the cones and the pedestal.

So

which is approximately 7.21 inches.

Then

So,

which is approximately 21.63 inches.

Therefore, b is approximately 14.42 inches.

Now, the pattern we want to lay out flat is cut from a circular disc of radius 21.63 inches and we remove a piece of a circular disc of radius 7.21 inches (the small cone). Like this:

What is left to determine is how big is the angle from A around to B to get the arc lengths and ?

In order to determine the angle we use the relation that the arc length S is given by

where r is the radius and the angle is in radians.  We can set this up using either the small circle or the large one.  Using the small circle with radius of 7.21 and arc length of , we have

Therefore,

Multiply by the conversion factor to get degree measure and you have:

This is a reflex angle slightly larger than a straight angle, as pictured above.

The same value for the angle can be found by using the arc length and the radius 21.63. Indeed, 21.63 = 3(7.21).

Now,  to make the pattern, draw a circle of radius 21.63 and a smaller circle of radius 7.21.   Mark off any radius, and then measure an angle of 200 degrees.   Cut out the blue figure and join the two sides together.

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