Equal Cores and Shells in Circles and Spheres
Equal Cores and Shells in Circles and Spheres
The area of the red circular ring between the radii and is given by . This is equal to the area of the blue disk of radius given by if the three radii satisfy +=. Visually, the equality of the areas of the shell and disk is often not very obvious, which might loosely be classed as an optical illusion.
r
2
r
3
A=π-
2
r
3
2
r
2
r
1
π
2
r
1
2
r
1
2
r
2
2
r
3
The 3D analog compares the volume of a red spherical shell and a blue central sphere. Recall that the volume of a sphere of radius is given by . The relation between radii is now given by +=. The spheres are shown in transparent hemispherical cross section.
r
V=π
4
3
3
r
3
r
1
3
r
2
3
r
3