While reading this chapter of the Feynman Lectures I came across a statement I didn't know how to prove.
He mentions below Eq. 4.30 that when you take a surface and tilt it by some angle $\theta$, the area of the new surface is increased by a factor of $\frac{1}{cos\theta}$.
Of course, simply changing the orientation of a surface does not change its area. What he means is that you cut the cone at a different angle and that the new surface is still sufficiently close to the original so you can use the same E. (They may for example share a vertex.)
I think this rule only works if the original surface is a spherical one like the ones mentioned above Eq. 4.30 although it's not said explicitly. And that it only works for infinitesimal surface areas too.
So my question is, how do I prove this? Tips, explanations and the proof itself are all welcome.
Thanks for reading.
It can be shown in a precise way but I will try to give the intuation behind this.
If you understand the reason for two dimensional object, it become more easy.
Now, $L_2$ is a line and $L_1$ is a projection of it to $x$ axes then you can say that,
$$|L_2|=|L_1|\dfrac{1}{cos(\theta)}$$
where $\theta$ the angle between the $L_2$ and $L_1$.
Now, instead of a line piece, let have a piece of plane $P_2$ and its projection to $x,y$ plane $P_1$.
It is a little bit hard to see but $$area(P_2)=area(P_1)\dfrac{1}{cos(\theta)} $$ where $\theta$ is the angles between the normal vectors of $P_1,P_2$.
The reason is simply this, only one dimensional property of $P_2$ changes. (When you look at your shadow it can be longer than you but width is same and $P_1$ is shodow of $P_2$ ) Thus, the ratio of areas are same the ratio of length.
Note : When you see that this equality for planes, then it become approximation for any other surfaces by tangent planes.