Getting both semi-axis of an ellipsoid by knowing its surface area.

2017-12-03 03:05:22

I know that the spheroid has a surface area, $A$, of $900cm^2$ and that its semi axis $a$ and $c$ are such that $c = 0.707a$. I wish to know the values of $a$ and $c$.

There's this formula which approximates the surface area of an ellipsoid (I'm not sure if its correct, I couldn't find it anywhere besides Wikipedia):

$$A = 4\pi(\frac{a^pb^p+a^pc^p+b^pc^p}{3})^{1/p}, p \approx 1.6075$$

In this case, since its an ellipsoid of revolution, $a = b$ and the expression can be simplified to:

$$A = 4\pi(\frac{a^pa^p+a^pc^p+a^pc^p}{3})^{1/p} = 4\pi(\frac{a^{2p}+2a^pc^p}{3})^{1/p}=4\pi(\frac{a^{2p}+2a^p(0.707a)^p}{3})^{1/p} \equiv$$

$$\equiv 3(\frac{A}{4\pi})^p=a^{2p}+2a^p(0.707)^p$$

It would be solved by "passing" everthing, except for the $a$ to the left ($a$ as a function of $A$) and injecting the values. Sadly, I do not know how to proceed from that last step, if it is even possible. But, if there's any other way of getting there, please point it out.

Thanks.

You ca

  • You can use a substitution with $h = a^p$ and you'll end up with a quadratic equation.

    2017-12-03 03:34:46