# rotating a circle around axis

When I imagine a circle($x^2+y^2=1$) and rotate it around the x axis, I get a shape of sphere.

So my question is why $(2πr)\timesπ$ and $(πr^2)\timesπ$ do not represent the area and volume of sphere.

I see that there are overlapped points especially in case of the volume, but still resulting values are too far away from the values obtained using spherical coordinates system.

What am I missing?

• For the surface area of the sphere, consider a point along the perimeter of the original circle. The distance that it travels around the axis of rotation depends on where the point originally is. It could be that it doesn't move at all during the rotation, or it could make a large circle. As you can intuitively see from this, you need to use integration to derive the surface area of the sphere. Jul 13 '18 at 6:24
• By the way, if you want to make a sphere from $x^2 + y^2 = 1$, maybe you want to rotate around the $z$-axis, no? Otherwise it doesn't change at all. Jul 13 '18 at 6:25 