hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.

The Math Fun Facts App!

List All : List Recent : List Popular
About Math Fun Facts / How to Use
Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Medium level:

# Surface Area of a Sphere

The area of a disk enclosed by a circle of radius R is Pi*R2.
The formula for the circumference of a circle of radius R is 2*Pi*R.
A simple calculus check reveals that the latter is the derivative of the former with respect to R.

Similarly, the volume of a ball enclosed by a sphere of radius R is (4/3)*Pi*R3.
And the formula for the surface area of a sphere of radius R is 4*Pi*R2.
And, you can check that the latter is the derivative of the former with respect to R.

Coincidence, or is there a reason?

Presentation Suggestions:
Let your students tell you those geometry formulas if they remember them.

The Math Behind the Fact:
Well, no, it is not a coincidence. For the ball, a small change in radius produces a change in volume of the ball which is equal to the volume of a spherical shell of radius R and thickness (delta R). The spherical shell's volume is thus approximately (surface area of the sphere)*(delta R). But the derivative is approximately the change in ball volume divided by (delta R), which is thus just (surface area of the sphere).

So, if I tell you the 4-dimensional "volume" of the 4-dimensional ball is (1/2)*Pi2*R4, what is 3-dimensional volume of its boundary?

See also Volume of a Ball in N Dimensions.

Su, Francis E., et al. "Surface Area of a Sphere." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    calculus
Subjects:    geometry, calculus, analysis
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
3.77
current
rating
Click to rate this Fun Fact...
*   Awesome! I totally dig it!
*   Fun enough to tell a friend!
*   Mildly interesting
*   Not really noteworthy
and see the most popular Facts!
Get the Math Fun Facts
iPhone App!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!