How to Develop a Cone - Cone Development
How to develop a cone or how to create a flat pattern of a cone can be achieved in a few easy geometrical steps. The geometrical method shown below does however have inaccuracy, so at the end of this hub I have included a mathematical formula to help produce an accurate conical development.
First of all you should draw your cone in elevation and plan as shown in Figure 1.
Divide your plan view up into equal segments. I have broken the plan view shown up into twelve (12) equal segments. (The accuracy of the development will increase with the number of segments that you break the plan view of the cone into.)
For twelve (12) equal divisions break the plan view up by drawing lines through the centre line at 30, 60 90 and 180 degrees.
We will use these divisions in a couple of steps but first we must lay out the initial overall cone development.
In Figure 2 I have shown the use of a drawing compass to get the true length of the cone side. You could use any computer aided drafting (CAD) method to obtain this length.
With Figure 3 as an example (shown below), take this true length of the cone side and scribe a full circle.
This is now your main development.
Now as in Figure 4 below, draw a horizontal line from the centre of this circle to intersect the outside diameter.
We now go back to our cone plan view and take one of the equal divisions we made earlier, as shown in Figure 5. You can do this using your compass or CAD program.
With this equal division on your compass and starting back at the horizontal line on your main development scribe the equal number of division. In my example I will scribe twelve (12) equal arcs along the circumference of the main development as shown in Figure 6.
Now finish this off by drawing a line from the last arc to the centre of the main development as shown in Figure 7.
An finally draw lines from every arc back to the center of the main development as shown in Figure 8. We can disregard the top half of the main development as the equal divisions form your full cone development.
Note: If this cone was to be cut out of steel these lines from the circumference to the center are actually called press lines.
You can check your work by printing out the development. Cut around the outline of the development and lightly folding each press line down. Join the two outside straight edges with sticky tape and there you have your cone!
Finally I promised a mathematical formula to help produce a more accurate cone development. The inaccuracy occurs when you use a compass to break your cone base circle up into equal segments. You can check this buy measuring one of your arc lengths and multiplying this by the number of divisions. Compare this result with the formula for the circumference of a circle which equals Pi times by diameter. You will notice the calculated result is longer.
Now to help produce your accurate cone development use this formula:
Angle = (D1 x 360) / D2
- Angle = the included angle between the outside lines of the main development
- D1 = the diameter is of the base of the cone (see diagram below)
- D2 = the diameter is of the developed cone, which you get from the elevation of your cone (see diagram below)
To use this formula, scribe your main development radius then draw your horizontal line. Then use the formula to calculate the included angle of the outside lines, grab a protractor (or use AutoCAD) and measure/draw this included angle out from the horizontal line.
The formula described in diagram...
Hope this helps, please drop me a comment if anything needs clarifying and I will certainly update this hub. Thanks!
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How to draw fast in AutoCAD, shows you how to improve your drafting productivity over time (by changing the way you enter commands) even with updates of AutoCAD where you get lost in the new user interface.
How to develop a Pyramid, follow similar easy steps on how to develop a flat pattern of a pyramid.
How to develop a truncated cone, follow similar steps on how to develop a truncated cone.
How to develop a Cylinder, follow easy steps to develop a flat pattern of a cylinder.