Understanding the Volume of a Sphere Formula [Using High School Geometry]
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone wants a visual proof of the sphere's volume or surface area via the pyramid decomposition, and when discussing why that argument works.
Recommended by 2 people across 2 different months, 2015 to 2024.
What it is
A short geometry explainer that derives the volume of a sphere, (4/3)\u03c0r^3, by slicing the sphere into many thin pyramids whose apexes meet at the center. It uses only high school geometry and algebra and is presented on the mathematicsonline channel.
What people said
Recommends Hrhm · 2016-06 · Mathematics Stack Exchange · on “Visualization of surface area of a sphere”
You might want to watch this video: [LINK] Also, if we are allowed to assume that the volume of a sphere is $\frac{4}{3}\pi r^3$, it is not hard to show that the surface area should be $\frac{3V}{r}$, thus proving that it is $4\pi r^2$. We can do this by splitting the sphere into many pyramids. See this video: [this video]
Recommends Nate · 2024-10 · Mathematics Stack Exchange · on “Proofs of the Volume of a Sphere.”
Here you go: [this video] Here's why the pyramids have a volume of Bh/3: [LINK] As a bonus, here's one for the surface area: 0:00 - 4:17 and 8:15 - 10:30 [LINK]
Cites user89167 · 2015-04 · Mathematics Stack Exchange · on “Why does this sphere volume proof work?”
See this video [this video] , but I can explain it without you watching it (nothing important is said and you can see the video in mute): Basically all it does is cut the sphere into 'infinitesimal' pyramids with bases on the surface area of the sphere and main vertices aligning in the centre of the sphere. The sum of the volumes of these pyramids simply turns out to be $\frac{4}{3}\pi r^3$. But this example [LINK] shows that such methods don't necessarily work. Why does it work in this case?