why is the area of the intersection of the cylinder and the plane Pi?

Question

I have a cylinder equation (x-1+az)^2+(y+bz)^2=1. The intersection of

the cylinder and the YOZ plane should be bigger than the base when it

is an ellipse. The base is the circle (x-1)^2+y^2=1 with area Pi. The

intersection is (az-1)^2+(y+bz)^2=1. I tried different a's and b's,

The area is always Pi, for example letting a=1 and b=10. The

calculation is done by using Wolfram|Alpha: Computational Knowledge Engine. The short axis of

this ellipse seems smaller than 2. I think since the

base circle has radius 1, the ellipse should have short axis 2.

The result is very counter intuitive. When one cuts the cylinder

with a plane and the result is an ellipse, the area of the ellipse

should be bigger than the base circle.

Re: why is the area of the intersection of the cylinder and the plane Pi?

I'd like to remove the second paragraph. It is not true for an oblique cylinder. I still want to know if there are some a and b to make the intersection bigger than Pi.

Re: why is the area of the intersection of the cylinder and the plane Pi?

I found it if the wolfram website is right. When a=.1 and b=.2, the area is about 10Pi.