About a short proof of Pizza Theorem

Question

I met the Pizza theorem today after this link activated several years later: Proof of the Pizza Theorem

Then I searched it with Google and found a very short proof made by calculus: https://web.maths.unsw.edu.au/~mikeh/webpapers/paper57.pdf

I understood the proof except the area formula. I have some scepticism. Why is the area formula $$\int_{\theta_0}^{\theta_1}\frac{1}{2}r(\theta)^2d\theta$$ is still valid although the point $P$ is not the center? Can somebody explain this? I couldn't.

Thanks before for your explanation.

Answer 1

$r(\theta)$ is the radius from $P$ to the outside part of the pizza. The area of a small segment of the pizza subtending an angle $\delta\theta$ is the area of a triangle, the height times the base times a half. The height is $r(\theta)$ and the base is $r(\theta)\times\delta\theta$ since the base's length is proportional to the angle times the radius. So you get $$\frac12 r(\theta)^2\delta\theta.$$

The rest is just taking the limit as $\delta\theta$ becomes infinitessimal.

There is a fuller discussion in this Wikipedia article.