I understand how the area of a pyramid is derived and that a cone is just a pyramid with a circular base.
But I can't help but to imagine the circumference being straightened and being a triangular prism. The circumference is $2\pi r$ and the area of the triangle is $\frac 12 r h$. $\frac 12 r h$ is being replicated $2\pi r$ times!
I think my mathematical intuiton is failing me because things like this keeps me up at night.
Following that, the area of the sphere. Damn Archemedis is also telling me that the volume of a sphere is a ratio to the imaginary cylinder it is inscribed in. But my stupid mathematical intuition tells me it is half the circumference times the area of the circle. Which can be written as $\frac 12 \pi r \times \pi r^2$. Purge this arrogance from me by explaining simply why my intuition is false without using the method of exaustion.