Equal Cores and Shells in Circles and Spheres

figures

circles

spheres

r

3

20

r

2

18

r

1

= 12.943

r

2

= 18

r

3

= 20

V=

4

3

π

3

r

3

-

3

r

2

=

4π

3

r

1

3

= 9081.297

The area of the red circular ring between the radii

r

2

and

r

3

is given by

A=π

2

r

3

-

2

r

2



. This is equal to the area of the blue disk of radius

r

1

given by

π

2

r

1

if the three radii satisfy

2

r

1

+

2

r

2

=

2

r

3

. 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.

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

r

is given by

V=

4

3

π

3

r

. The relation between radii is now given by

3

r

1

+

3

r

2

=

3

r

3

. The spheres are shown in transparent hemispherical cross section.