# Thread: Expression of Formula In Some Terms

1. ## Expression of Formula In Some Terms

The formula for the area of a spherical cap with radius (the sphere's radius) $R$ and height $h$ is given as

$A = 2{\pi}Rh$

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

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

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

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

2. Originally Posted by fishcake

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

$(R-h)^2 + r^2 = R^2$
$(R-h)^2 = R^2 - r^2$
For $r$:

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

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

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

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

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

For $R$:

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

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

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

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

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

$R = \frac{ r^2+h^2}{2h}$

3. Originally Posted by fishcake

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

For $R^2$:

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

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

$\frac{r}{R} = \frac{\sqrt{2Rh-h^2}}{\frac{ r^2+h^2}{2h}} = \dots$

4. Hmm, I'm still confused here. How do I make use of $\frac{r}{R}$ and $R^2$ inside the formula for $A$? What does "express $A$ in terms of $\frac{r}{R}$ and $R^2$" actually means anyway? Maybe I have misinterpreted the question, which can be found here. It's question 1(b).