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I am trying to do an aerodynamic drag equation on a descending parachute (the round variety) and have no idea what the reference area on one would be. I know for a sphere, you can use radius*radius*PI to get the reference area. Is that the same for a parachute?

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Poking around on Google with various search terms that included "parachute shape" i came upon "The Parachute Manual" by Dan Poynter. Table 8.1.7 from that book catalogs empirical data on a host of parachute shapes. Assuming that the parachute is round, as you say, and does not have any holes (apparently, many designs purposefully include gaps to improve stability), the first section of that table (on page 457) is the one to look at. To interpret the terms of the table, I tried reading the glossary. It gives the following three definitions:

  1. nominal diameter: the diameter of a circle made with the same amount of cloth as the parachute.
  2. projected diameter: the diameter of the real parachute when it is inflated.
  3. constructed diameter: the diameter of the real parachute when it is not inflated.

From this it is clear that the table says "nominal diameter" but means "constructed diameter." The table is in fact using a nominal diameter of 1 for all the parachutes it lists.

Using this correction, the table reveals that the ratio between projected diameter and nominal diameter for round parachutes varies between 0.6 and 0.7 This means that given the projected radius, the area is, roughly speaking, somewhere between $2\pi r^2$ and $3\pi r^2$.

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