<story><title>But why is a sphere&apos;s surface area four times its shadow? [video]</title><url>https://www.youtube.com/watch?v=GNcFjFmqEc8</url></story><parent_chain><item><author>Herodotus38</author><text>Does this relationship generalize? Hypothetically carrying further, if you take the derivative of the area (wrt radius) you get 2pi.  A one dimensional circle is a line segment of length 2*r right? So how is 2 pi related to a line segment? Is it in some way analogous to the way perimeter and area are for 2d or surface area and volume?  I don&amp;#x27;t know the answer.</text></item><item><author>azernik</author><text>Fun fact: by the same logic, the perimeter of a circle is the derivative of its area.</text></item><item><author>war1025</author><text>It maybe answers a different question, but I always thought it was neat that the surface area for a sphere is just the derivative of its volume. Beyond that, I guess I never thought about it much.</text></item></parent_chain><comment><author>thaumasiotes</author><text>&amp;gt; Does this relationship generalize?&lt;p&gt;Yes, the fundamental theorem of calculus guarantees this will be true. It&amp;#x27;s the same phenomenon as that if your &amp;quot;volume&amp;quot; is the definite integral of f(t) from fixed &lt;i&gt;a&lt;/i&gt; to variable &lt;i&gt;x&lt;/i&gt;, then the rate of change of the volume (= the surface area) is f(&lt;i&gt;x&lt;/i&gt;), the length of the infinitesimal sliver you&amp;#x27;re about to add to the volume.&lt;p&gt;You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres. Each layer contributes volume to the solid sphere equal to its own area. (And this is what it means to calculate the volume of a sphere as the integral of the area of a spherical shell as the radius goes from 0 to r.)</text></comment>
<story><title>But why is a sphere&apos;s surface area four times its shadow? [video]</title><url>https://www.youtube.com/watch?v=GNcFjFmqEc8</url></story><parent_chain><item><author>Herodotus38</author><text>Does this relationship generalize? Hypothetically carrying further, if you take the derivative of the area (wrt radius) you get 2pi.  A one dimensional circle is a line segment of length 2*r right? So how is 2 pi related to a line segment? Is it in some way analogous to the way perimeter and area are for 2d or surface area and volume?  I don&amp;#x27;t know the answer.</text></item><item><author>azernik</author><text>Fun fact: by the same logic, the perimeter of a circle is the derivative of its area.</text></item><item><author>war1025</author><text>It maybe answers a different question, but I always thought it was neat that the surface area for a sphere is just the derivative of its volume. Beyond that, I guess I never thought about it much.</text></item></parent_chain><comment><author>baddox</author><text>Seems to me that a one-dimensional circle would be two points on a number line separated by distance 2*r.</text></comment>