]> Surface Area of a Sears-Haack Body
PDAS Home > Contents > Difficult Problems > Sears-Haack Area
Public Domain Aeronautical Software (PDAS)

If the mathematical expressions do not display properly on your browser, you may view the page in PDF. I have a page to help you get started with MathML.

This question comes up from time to time. It seems quite straightforward. To compute the surface area of a body of revolution, you must integrate the function

$S= 2π∫r1+ ( dr dx )2 dx$

from nose to tail.

For a Sears-Haack body (fixed length and volume)

$r= [4x (1-x )] 3 4$ $dr dx =3 [4x (1-x )]- 1 4 (1-2x)$

where r is radius / max.radius and x is the distance from nose / body length.

Now, the only problem with this is that no one seems to be able to carry out this integration symbolically. This includes the powerful symbolic math aids. However, it is rather straightforward to perform the integration numerically and I have a numerics page that describes the mathematical and programming techniques. For those of who who just want the answer, get the factor from the following table for the fineness ratio (length/max diameter) of your body.

ffactor
20.7811
40.7358
60.7265
80.7232
100.7217
120.7208
140.7203
160.7200
180.7197
200.7196
220.7195
240.7194
260.7193
280.7192
300.7192
400.7191
500.7190

Then compute the surface area from

$S=(2π rmax L)factor$

The quantity in parentheses is the surface area of a cylinder of the same length and diameter as the Sears-Haack body. So, if you just remember that the area is about 72 percent of that, that will be good enough for most purposes.

PDAS Home > Contents > Difficult Problems > Sears-Haack Area
Public Domain Aeronautical Software (PDAS)