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Circles and Spheres - A neat observation

Circles and Spheres - A neat observation

I was thinking about how the surface of an object could be related to its interior.

Let's start simple. Consider a circle of circumference c and radious r. Its "interior" is the area and its "surface" is the circumference.

So let's take the ratio of the two.. area/circumference

The area of the circle is cr/2 so dividing this by c gives us r/2 for the ratio.

For a sphere the ratio would be volume/(surface area). The volume is (4/3)(pi)r^3 and the surface area is 4(pi)r^2 and so we get r/3 for the ratio.

Interesting. A circle is a 2D object and the ratio is r/2. A sphere is a 3D object and the ratio is r/3.

So let's make the conjecture that for an n-dimensional sphere of radius r the ratio would be r/n. True or false?

Content written and posted by Ken Abbott

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