Considering a Sphere
The article explores a geometric visualization to understand the volume of a sphere by relating it to the surface area and using concepts like Cavalieri's principle and decomposition into infinitesimal cones. It begins with a cube divided into square-based pyramids to derive the formula for the volume of a cone, then extends the reasoning to spheres. By connecting the volume of each small 'cone' from the sphere's surface to its center, the author derives the volume of the sphere in terms of its surface area.
Opening excerpt (first ~120 words) tap to expand
Starting with a circle ... Recently someone posted an animation of a way to visualise the area of a circle. It was on Twitter, so I can't find it again, but here at right is an image taken from http://mathandmultimedia.com/ Someone asked if there was a similar visualisation of the volume of a sphere, or the surface area of a sphere, and that started me thinking. We know that for a circle, for some reason, the circumference is the derivative of the area, and similarly for the sphere, the surface area is the derivative of the volume, and I wondering about visualisations of that, and related things. This is what I came up with - I'd welcome comment, thoughts, and feedback. It's worth noting that in places here I'm playing fast and loose with infinities and limits.
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Excerpt limited to ~120 words for fair-use compliance. The full article is at Co.
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