# Math Help

can someone explain why the formula for the surface area of a shape is the derivative of the volume?
I’ve noticed this with a few shapes and was just wondering why
Original post by lila__
can someone explain why the formula for the surface area of a shape is the derivative of the volume?
I’ve noticed this with a few shapes and was just wondering why

I guess youre referring to something like spheres, or circles where the perimenter is the derivative of the area? You can imagine a sphere's volume split into cones originating from the centre and each cone has the same infinitesimal base area and the sum/integral is the surface area of the sphere. So the volume is "constant height" cones and youre integrating over the surface area. Its similar to the pizza slice argument for the area / perimeter of a circle.
(edited 2 months ago)
Original post by lila__
can someone explain why the formula for the surface area of a shape is the derivative of the volume?
I’ve noticed this with a few shapes and was just wondering why

Nice observation!

The intuition actually works better backwards - i.e. the volume is the integral of (cross-sectional) area, if you think about integrals as "adding a bunch of little things". Then the derivative - being the "anti-anti-derivative", do just as what you observed.

In fact, if you happen to know the cross sectional area of an object as a function of its height, the volume is exactly the integral of the function over the height range. (And that is how the disc method is presented in books - adding a bunch of areas of circular discs with the radii being the function at various points.)
(edited 2 months ago)