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### Deformed Bullet

```Date: 04/11/2002 at 12:03:14
From: Allison Milam
Subject: Circle/Ellipse Diameter

If you initially have a circle and it gets smushed into an ellipse,
how do you determine what the diameter of the initial circle was,
knowing the major and minor diameter of the ellipse?
```

```
Date: 04/11/2002 at 12:54:44
From: Doctor Peterson
Subject: Re: Circle/Ellipse Diameter

Hi, Allison.

That depends on how it was "smushed"! What was preserved - does it
still have the same area (as if it is made of clay or some
uncompressible material), or the same perimeter (as if it is contained
in an unstretchable membrane, but the inside can compress), or what?

question. You may need to use our Geometric Formulas page to see how
to carry out the plan:

Ellipse
http://mathforum.org/dr.math/faq/formulas/faq.ellipse.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/11/2002 at 13:25:07
From: Allison Milam
Subject: Circle/Ellipse Diameter

I am a forensic scientist in the field of firearm and toolmark
examination. We use the diameter of the base of a bullet as a way of
figuring out caliber. Therefore, when we get a damaged bullet where
the base has changed from a circle to an ellipse we would like to be
able to determine what the initial diameter was.
```

```
Date: 04/11/2002 at 13:33:54
From: Doctor Peterson
Subject: Re: Circle/Ellipse Diameter

Hi, Allison.

You can probably assume that the area remains the same, although I
could easily imagine metal deforming in such a way that the area might
change. With that assumption, you just have to find the area of the
ellipse

K = pi A/2 B/2

where A and B are the major and minor axes (diameters), as opposed to
the usual a and b, the semi-axes (radii).

Since a circle of diameter D with the same area would require

K = pi (D/2)^2

we have

pi D^2/4 = pi AB/4

D^2 = AB

D = sqrt(AB)

So just take the square root of the product of A and B (called the
geometric mean) as your presumed original diameter.

I would recommend testing this on some deformed bullets whose caliber
you know, in order to check our assumptions.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/11/2002 at 13:49:04
From: Allison Milam
Subject: Circle/Ellipse Diameter

Thank you very much for your time.  We are going to manually deform
some known caliber bullets with a hammer and try out these formulas.
If it works I will write an article to share the information with
others in the field.  Again, thank you for your help.
```
Associated Topics:
High School Conic Sections/Circles

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