Making frustums doesn't have to be frustrating

Figure 1
Roof flashing for vent pipes, outlets for leaders, conductor pipes and tapering elbows are just a few of the many sheet-metal pieces that can be made using radial lines. From basic cones to the more complicated frustums and irregular cone sections, using radial lines can be a time-saving alternative to the sometimes tedious method of triangulation and complex math calculations. As difficult as it may seem, radial-line development is a straightforward procedure and fairly easy once you learn a few basic steps.

There are different methods for developing the patterns of these fittings; choosing whether the cone should be developed using radial lines or triangulation would depend much on the pitch or taper of the cone. The larger tapered cones are easily made using radial lines. As the taper becomes larger, the apex of the cone becomes closer to the drawing area. The smaller the taper, the further the apex becomes from the working area where the pattern is to be made. On the shorter tapers, especially in the larger diameter frustums, it would be more practical to use triangulation. Figure 1 shows the basic cone (A), frustum of a cone (B), truncated cone (C) and irregular section of a cone (D).

Figure 2

## Making a cone's frustum by radial line

Figure 2 illustrates a basic frustum of a cone and the development of the frustum, as shown by the shaded area of the cone and the shaded area of the half pattern. Two things are needed to develop cones using the radial-line method: An elevation view and a plan view. An elevation view would include the projection lines, which establishes the apex. A plan view of the cone, also known as a profile view, allows you to divide the cone into equal spaces and project them to the base of the cone, where you would use them to develop the pattern.

Figure 3

## Establishing arcs with trammel points

Set the trammel points the distance of the apex to A. Now using the apex as a center, make an arc from A to A'. Now reset the trammel points to the distance from the apex to B. Again using the apex as center, strike an arc from B to B'. From the apex, draw a straight line as shown in Figure 2 to 1'. This line establishes the beginning points for which the pattern can be developed. This line can be drawn at any distance away from the elevation view. The distance 1' to B' is to equal that of the circumference of the base.

Once B' has been established, draw a line from B' to the apex. This will automatically establish Point A'. If you do this, Point a to A' will be equal to the top circumference of the frustum.

Frustum of a cone with a pitch (Step No. 1 - The elevation view)

Figure 3 illustrates how to deal with a cone on a pitch or a base with an angle other than 180 degrees. Draw the elevation of the complete cone and include a plan view, as shown in the previous drawing. Divide the plan into equal spaces. The plan in Figure 3 was divided into 12 equal spaces. From the base line, create perpendicular lines to each one of the points in plan view, as shown.

Since the two halves (top and bottom of the plan view) are symmetrical, it is only necessary to project these lines to the top half. The bottom of the plan is marked off and numbered (more on this later). From the point where each line projected from the plan meets the base line of the cone, draw a line to the apex. Draw the line in the elevation of the desired angle as shown by the line that connects points a, b, c, d and e. It makes no difference what the angle is; it all works out the same.

Where the line drawn on the angle intersects with the vertical lines drawn from the base to the apex, label each as shown by a, b, c, d, e, etc. To develop the pattern, you need to know the true length of the lines from the apex to a, b, c, d, e, f (f is not shown) and 7".

There are only two ways to find out the true length of lines: by looking perpendicular to its plane (as when looking at the plan view) and by looking at the end view of the plane (as shown at the elevation view B to 7" and A to a, which are shown as true-length lines.

You need to turn the whole cone so that each of the points, b, c, d, e and f, are on the same plane as B to 7". To accomplish the same thing, we can project lines that run parallel to the base of the cone to the same plane as B7" is on. These are marked as 1", 2", 3", 4", 5", 6" and 7". Line AB has already been established as the height of the fitting.

Figure 4

## Step No. 2 - Creating the pattern

Now look at Figure 4. The next step is to draw an arc that represents the circumference of the base. Set the trammel points at the distance of the apex to 1". Using the apex as center, make an arc at an extended length. Draw a straight line from the apex to Point 1.

After setting the trammel points or dividers at the distance used to divide the circumference of the plan view in Figure 3, strike off this distance from 1 to 2, 2 to 3, 3 to 4, etc., until you get back down to Point 1 again. You should have 12 divisions along the arc created from Point 1".

From the last division, draw a line to the apex. From the apex, draw a line to each of these reference marks along the arc struck from 1", as shown. Now, to establish the pattern of the cone, create a series of arcs using the apex as center from 2", 3", 4", 5", 6", 7" and Point B. Then, as shown with the red arrows, follow the arc from Point 1" over to the first line drawn from the apex to Point 1. This is the beginning point for creating the actual pattern of the frustum.

From Point 2", follow the arc to the intersection of the line that is drawn in from the apex to Point 2 on the base line. See the red arrows for the point of intersection.

From Point 3", follow the arc to the intersection of the line that is drawn in from the apex to Point 3 on the base line. Repeat these steps with points 4", 5", 6" and 7'.

The intersection of the arc from 7", and the line from the apex to 7 on the base completes half of the frustum. To complete the pattern for the second half, again follow the arcs drawn from points 7", 6", 5", 4", 3" and 2", until they intersect with the lines drawn from the apex to points 6', 5', 4', 3' and 2' on the base.

Figure 5

## Using radial lines to create the pattern for conical-shaped items

This pattern could be used to make gutters or eave troughs. Look at Figure 5. There's an elevation drawing and a plan view (half profile). The item is pitched; you have two slant lines from the elevation drawing, which allows you to extend these slant lines and generate the point of the apex.

Divide the plan or profile into equal spaces as shown and extend them to the base line of the cone, shown with the slashed red lines. Remember that the cone is the complete triangle including the apex. From the base of the cone, extend these lines to the apex, as shown with the solid red lines. Where these red lines intersect, the elevation view (intersecting planes) is where you should make your projection lines - parallel with the base of the cone - to the outer edge (slant line), establishing the true lengths to complete the arcs, as done in Figure 4.

There are many patterns that you will be able to take advantage of using radial-line development; however, it will depend on your ability to recognize radial-line forms. Be creative as you approach these different articles. The apex doesn't necessarily have to be on top. Sometimes you have to think upside down.

Look again at the cones in Figure 1. Without the plan or profile view, the cones could easily be a pyramid, and pyramids can be made using radial lines. One rule to remember is that the taper must remain constant from the apex to all points on the circumference of the base. This is why the base is included in all radial-line work. Once you do this, you can create any irregular form as long as the surface shares that of the cone.

A printable worksheet and more information on radial-line development can be found online.

(For more information on manual sheet-metal layout, write to The Sheet Metal Shop Resource Center, 516 Chicago Ave. Waukesha, WI 51388.)