Please tell me where my reasoning goes wrong:
Using multivariable calculus and other methods, one can easily show that the surface area of a sphere is equal to $4\pi r^2$ and I will consider this a fact. Now let's imagine a sphere with radius $R$. Let's center it at the origin of the standard 3-dimensional space so that its top point is located at $(0,0,r)$ and its bottom point is centered at $(0,0,-r)$. Now we will remove the bottom hemisphere and let the top hemisphere stay. The surface area of the top part is equal to half the whole sphere so it is equal to $2\pi r^2$. Now consider the point $(0,0,-r)$ and call it $P$. It is clear that every point on the top hemisphere can be connected using straight lines to our point. All points of the top hemisphere pass through the equator of our sphere which is a circle centered at the origin defined by $x^2+y^2=r^2$ which has area $\pi r^2$. All points on the hemisphere correspond bijectively to points on the circle. This implies that there are equal points on the sphere and our circle which implies that the hemisphere has surface area $\pi r^2$. But we know that the surface area of the hemisphere is $2\pi r^2$. This is only true if and only if $2=1$.
This is clearly wrong, where have I gone wrong? I think it has to do something with an infinite set being equal to its subset or something. I think an area is nothing more than the sum of infinitely many one-dimensional points and if two figures have equal points, their area should be equal.