A thin surface band has circumference times slant width, not circumference times horizontal width.
WHAT HAPPENS IF…Why does the square-root factor appear?
Interactive exploration
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Connect the picture to the ideaS=2π∫0bf(x)1+[f′(x)]2dx
With f(x)=x≥0, each band has circumference 2πx and slant differential √2 dx. Integrating gives πb²√2, equal to the cone formula πrℓ with r=b and ℓ=b√2. The base disk is a separate surface.
A detail that matters
Surface area is measured in square units, while volume uses cubic units.