Ratio of the Surface Area of a Sphere to a Cylinder

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Of all the shapes, a sphere has the smallest surface area for a given volume. What about a cylinder's surface area? With a properly chosen ratio of height to radius, how close can the cylinder's surface area get to the sphere's surface area of the same volume? Use the sliders to explore these questions without calculus. The bigger the ratio , the closer you are to a cylinder with the smallest surface for a given volume.


With equal volumes of the cylinder and sphere, define the parameter , where and are the height and radius of the cylinder. As can be calculated, the cylinder with the smallest surface area occurs for ; that is, when the diameter of the cylinder is equal to the height of the cylinder.


Contributed by: Jan Fiala (October 2014)
Open content licensed under CC BY-NC-SA



The control sets the volume and scale of the plot on the right. The control determines the proportions of the cylinder (large gives a long cylinder and small gives a short cylinder). The controls at the bottom are convenient for better visualization.


[1] J. Fiala. My Edited Video [Video]. (Aug 25, 2014) www.youtube.com/watch?v=tEGWreii6dM.

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