But why is a sphere's surface area four times its shadow?
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when they want an intuitive, geometric explanation of the 4πR² surface area of a sphere, including the ribbon-and-cylinder argument and its link to integration.
Recommended by 7 people across 8 different months, 2018 to 2024.
What it is
A 3Blue1Brown explainer that shows why a sphere's surface area is exactly four times the area of its shadow, building on Archimedes' idea of comparing a sphere to its enclosing cylinder. Grant Sanderson walks through animated geometric arguments, including slicing the sphere into ribbons and the cylinder projection, with a calculus-free intuition.
What people said
Recommends J.G. · 2018-12 · Mathematics Stack Exchange · on “Probability density function of $X$ with a uniform distribution on a unit sphere”
The easiest solution returns to the sphere's geometry. The part of the surface ranging from $x$ to $x+dx$ is a ribbon of radius $\sqrt{1-x^2}$ and thickness $d\arcsin x=\frac{dx}{\sqrt{1-x^2}}$ , so its area is $2\pi dx$ . This can be used to prove the classical fact that a sphere's area matches that of a cylinder just long and wide enough to hold it, because each ribbon on the sphere's surface has the same area as its shadow on the cylinder. 3Blue1Brown [LINK] has a great video on it here [this video] .
Recommends J.G. · 2019-07 · Mathematics Stack Exchange · on “Why is the area of a sphere is coming out as $ \pi^2R^2 $ instead of $4\pi R^2$?”
Since $x^2+y^2=R^2$ , $$xdx+ydy=0\implies\frac{dy}{dx}=-\frac{x}{y}\implies\frac{ds}{dx}=\sqrt{1+\left(\frac{dy}{dx}\right)^2}=\frac{R}{y}.$$ The surface is the integral of $2\pi yds$ , not $2\pi y dx$ . For an intuitive understanding of why, see this [this video] video's comparison of the sphere's surface to a cylinder's. (It'll even help you compare them without ordinary calculus.) Since $yds=Rdx$ , the final result is $$2\pi R\int_{x=-R}^{x=R}dx=4\pi R^2.$$
Recommends Vincent · 2019-12 · Mathematics Stack Exchange · on “Expected distance from the span”
This special property is *really really* special and doesn't hold for e.g. the circle. See this nice video about the property (that phrases it a bit differently): [this video]
Recommends Eric Towers · 2020-04 · Mathematics Stack Exchange · on “What does "surface area of a sphere" actually mean (in terms of elementary school mathematics)?”
You might benefit from this (and, less directly other) 3Blue1Brown videos: But why is a sphere's surface area four times its shadow? [this video] .
Recommends jackpipe · 2021-06 · Mathematics Stack Exchange · on “Intuitive proof of surface area of a sphere in relation to area of great circle using shadow.”
I'm trying to show my kids intuitive proofs of common formulas, and watched this excellent video from 3Blue1Brown: But why is a sphere's surface area four times its shadow? [this video] However, I didn't find the 'shadow' explanation intuitive (or I misunderstood it), and came up with an alternative, though I'm quite sure it's been thought before. Take a sphere and cut it and half leaving a dome, and illuminate it from above. Split the dome into equal horizontal rings, and note that each ring shades the 'ground' using some percentage of it's surface. For the ring at the top of the dome, the shadow will be 100% of the surface of the ring, whereas for the ring at the ground the shadow will be 0% of
Recommends Abolfazl Chaman motlagh · 2023-02 · Mathematics Stack Exchange · on “Calculating the surface area of a spherical cap using cylindrical coordinates”
your calculations are not wrong. since there are too many good references and illustrations on this problem, i'm gonna put some references here. this video is very good illustration of this problem [this video] you can follow Wikipedia [LINK] calculation on this problem This question [LINK] is very related
Recommends Mr Rubix · 2023-11 · Mathematics Stack Exchange · on “Surface Area is the derivative of Volume, what is the derivative of Surface Area?”
I think you are trying to explain that it represent the perimeter(Boundary) of the 2d object required to wrap completely the surface area(Boundary) of a 3d object. 8πr indicate that the Surface Area of a spere can be wrapped by 4 circles of perimeter "r" = 4*2πr : This has been proved in this video [this video] I sadly don't have enough reputation to provide a long answer so if you could add this yo your answer it would be awesome :) Have an awesome day. Dan
Recommends Mr Rubix · 2023-11 · Mathematics Stack Exchange · on “Surface Area is the derivative of Volume, what is the derivative of Surface Area?”
I see one obvious gemetrical signification as it represent the perimeter(Boundary) of the 2d object required to wrap completely the surface area(Boundary) of a 3d object. 8πr indicate that the Surface Area of a spere can be wrapped by 4 circles of perimeter "r" = 4*2πr : This has been proved in this video [this video] I sadly don't have enough reputation to provide a long answer so if you could add this yo your answer it would be awesome :) Have an awesome day. Dan
Recommends Nate · 2024-10 · Mathematics Stack Exchange · on “Proofs of the Volume of a Sphere.”
Here you go: [LINK] 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 [this video]
Cites Himanshu · 2020-02 · Mathematics Stack Exchange · on “Equality of Surface area of sphere and Cylinder”
I seen a video that shows How the surface area of sphere of radius R is equals to the surface area of cylinder of radius R and height 2 R . enter link description here [this video] It was an geometrical proof and I understand it but I want to proove it in calculas way by intergral.It mean I want to show why when you take area element in sphere or in cylinder they give the same inegtral result? How these integral area and area element related to each other ie. what kind of transformation from one variable to other take you from one perception to other? So Please Help me.