Again, as in the parallel line development, we must first identify certain components of our shape. Given the plan (base) and elevation (side) of the round pipe, we can geometrically construct the two-dimensional expanded pattern (surface). Essentially we are "unrolling" the round taper from three-dimensions into two-dimensions.
First, make sure the plan sits directly below the elevation and label points A and B on each. Note these points represent the diameter of the base, and are on either side of the elevation. Point O is labeled as the vertex of our cone, although it does not actually lie on the round taper.
Next, we must determine the base of our expanded pattern. So, draw new line OA to represent one side of the expanded pattern. Construct the two circles, each with center O, to represent the top and bottom of the round taper. Essentially, the curved surface of the round taper will be projected around these circles to find the complete expanded pattern.
So how far around the circumference of the longer circle will our expanded pattern occupy? Let d represent the diameter of our round taper (i.e. d is the length of segment AB). Let D represent the diameter of our largest circle (i.e. D is 2 times the length of radius OA). Then, we can create a ratio to determine how many degrees our entire expanded pattern should be (x).
Now using our side lengths, we can find x!
So we have found the full expanded pattern should be 169.58 degrees around the circle from segment OA.
We can see that points A and B are on opposite sides of the circumference of our shape. Therefore, point B will be in the center of our expanded pattern and we can adjust our ratio to locate point B on our expanded pattern.
So point B will be located 84.79 degrees around our expanded pattern from segment OA.
Label the points A and B on the base of our expanded pattern as shown below. Why are there two points A on the expanded pattern? Well, remember the expanded pattern has been unrolled. So when reassembled, this pattern will be our three-dimensional round taper with a seam at point A. So, basically, these points A are the same.
Finally, we can remove our construction lines and connect our points to see the completed expanded pattern. So, if you were to roll our expanded pattern back in to our round taper shape, you would have a taper with the size of the given plan and given elevation!
Click here to see the Geometer's Sketchpad file that generates the pattern for the round taper.
Click here to view the Graphing
Calculator file of the three-dimensional round taper.
Return to Essay 2: Complete the Pattern
