# Expression of Formula In Some Terms

• Feb 22nd 2010, 11:16 PM
fishcake
Expression of Formula In Some Terms
The formula for the area of a spherical cap with radius (the sphere's radius) $\displaystyle R$ and height $\displaystyle h$ is given as

$\displaystyle A = 2{\pi}Rh$

This formula can be expressed in terms of $\displaystyle R$ and $\displaystyle r$, where $\displaystyle r$ is the radius of the base of the spherical cap:

$\displaystyle (R-h)^2 + r^2 = R^2$
$\displaystyle (R-h)^2 = R^2 - r^2$
$\displaystyle R-h = \sqrt{R^2 - r^2}$
$\displaystyle h = R - \sqrt{R^2 - r^2}$

$\displaystyle A = 2{\pi}R(R - \sqrt{R^2 - r^2})$

Now, how do I express the formula in terms of $\displaystyle R^2$ and $\displaystyle \frac{r}{R}$? I've been trying for quite some time but I'm stuck...
• Feb 23rd 2010, 12:28 AM
pickslides
Quote:

Originally Posted by fishcake

This formula can be expressed in terms of $\displaystyle R$ and $\displaystyle r$, where $\displaystyle r$ is the radius of the base of the spherical cap:

$\displaystyle (R-h)^2 + r^2 = R^2$
$\displaystyle (R-h)^2 = R^2 - r^2$

For $\displaystyle r$:

$\displaystyle (R-h)^2 = R^2 - r^2$

$\displaystyle R^2-2Rh+h^2 = R^2 - r^2$

$\displaystyle -2Rh+h^2 = - r^2$

$\displaystyle 2Rh-h^2 = r^2$

$\displaystyle \sqrt{2Rh-h^2} = r$

For $\displaystyle R$:

$\displaystyle (R-h)^2 = R^2 - r^2$

$\displaystyle R^2-2Rh+h^2 = R^2 - r^2$

$\displaystyle -2Rh+h^2 = - r^2$

$\displaystyle 2Rh-h^2 = r^2$

$\displaystyle 2Rh = r^2+h^2$

$\displaystyle R = \frac{ r^2+h^2}{2h}$
• Feb 23rd 2010, 12:31 AM
pickslides
Quote:

Originally Posted by fishcake

Now, how do I express the formula in terms of $\displaystyle R^2$ and $\displaystyle \frac{r}{R}$? I've been trying for quite some time but I'm stuck...

For $\displaystyle R^2$:

$\displaystyle R = \frac{ r^2+h^2}{2h} \implies R^2 = \left(\frac{ r^2+h^2}{2h}\right)^2 =\dots$

For $\displaystyle \frac{r}{R}$:

$\displaystyle \frac{r}{R} = \frac{\sqrt{2Rh-h^2}}{\frac{ r^2+h^2}{2h}} = \dots$
• Feb 23rd 2010, 10:42 PM
fishcake
Hmm, I'm still confused here. How do I make use of $\displaystyle \frac{r}{R}$ and $\displaystyle R^2$ inside the formula for $\displaystyle A$? What does "express $\displaystyle A$ in terms of $\displaystyle \frac{r}{R}$ and $\displaystyle R^2$" actually means anyway? Maybe I have misinterpreted the question, which can be found here. It's question 1(b).