How to Develop a Truncated Cone

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Follow these instructions below to develop a truncated cone. A truncated cone is a conical shape which has had it's top cut off at an angle. Sometimes you might need to develop these truncated cones as duct elements. Developing a truncated cone can be difficult but as long as you understand the concept of 'True Length Line' then this development will be easy for you.

Firstly begin by drawing your truncated cone elevation as shown above. Once you have your elevation you should draw your plan view (or top view). I have produced my plan view underneath the elevation.

You need to divide your plan view up into equal segments and then transfer these points to the elevation (as I have shown above).

Now you need to draw your end elevation (or side view) of the truncated cone. You need to do this to help produce your points that will assist in your truncated cone development. I have started my end elevation in the drawing above.

Once again you need to divide your end elevation up into equal distances, the same equal distances that you divided your elevation into (as shown above).

As per the detail above you now need to project the intersection points of these divisions to your truncated cut edge across to your end elevation and down (or up) to your plan view. These projected lines need to intersect with the corresponding divisions in the plan and end elevation.

In the end elevation I have drawn in the oval shape of the truncation.

From the end elevation you can project the two outside points of the plan view truncation. To do this you can use a 45 degree line like I have shown above.

In the diagram above the compass is set along the 'True Length Line' edge. In the diagram below I have projected this edge to the tip of the cone (if the cone were a complete shape).

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Now this is the tricky part of truncated cone development. You need to project the top of the truncated line in the plan view to the centre line of the plan view. This line is then projected up (or down) to the true length line in the elevation. I have shown the true length line in red in the elevation. Now using your compass take this true length line and scrib an arc onto your development.

I have shown this method below for all the divisions of the plan view to generate each true length line. If you scroll down to the development you well see that I have scribed every true length line below. (If you are having trouble understand me with this particular instruction I'll try to develop this article and explain better - like a work in progress, please send me an email or drop a comment below)

Below its the generation of the development. Start by creating a circle from the distance of the apex of the cone to the base along the true length line (or the edge of the cone in the elevation view). Then breack up the circle into equal parts, the same equal parts that you divided your plan view into. See below:

Below I have just tidied up the view by drawing lines from the division to the apex of the cone.

Below is showing how you transfer your true length lines to these cone divisions, with a cone you can use the same true length line twice around the centre line of the cone (centre line is shown darker)

Draw a curve between these true length line arcs, as shown below, now you can tidy up the development and you will have the development of a truncated cone.

Below are two developments of a truncated cone, both are correct. One shows the develoment join with the longest edge and one shows the development join with the shortest edge. If you are welding steel to make a duct element then you should use and start your development with the shortest edge as this is the most economical method of fabrication.

Rose on January 26, 2020:

Its very nice, it has really help me a lot keep it up.

Stban Worestee on April 11, 2015:

How do you find the volume of such figure?

LeonJane (author) from Australia on February 25, 2014:

Yes it is.

I have my plan view under the front view (elevation), sorry for the confusion. To be correct it should be above the elevation, but the development will still be able to be produced.

arindam maji on April 23, 2013:

what will be the surface if cut by section plane inclined at 30degree to horizontal plane

john on January 04, 2013:

hi there,your methods are great and easy to understand.just checking if you have any formulas for developing rectangular shape to round.

regards

Emma from Houston TX on November 09, 2012:

Wow! that's good hub. Keep it up.

Tebaiti on September 26, 2012:

super super