# 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