Expert: Paul Klarreich - 7/4/2007

Question
A doughnut is big enough to be fitted perfectly in a rectangular box of dimensions 4 in x 4 in x 1 in. The taste doesn't come from the size, but from the fact that the doughnut surface is sprinkled with 7 grams of coconut per square inch. Find how much coconut was used to prepare this tasty treat.

Answer
Questioner:   Angela
Category:  Calculus
Question:  A doughnut is big enough to be fitted perfectly in a rectangular box of dimensions 4 in x 4 in x 1 in. The taste doesn't come from the size, but from the fact that the doughnut surface is sprinkled with 7 grams of coconut per square inch. Find how much coconut was used to prepare this tasty treat.
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Hi, Angela,

You want the surface area of a torus.  The parameters are apparently:

1. The center of the torus is at the origin.
2. The radius of the circular cross-section is r = 0.5.
3. The 'ring' is a circle with radius, R = 1.5.

I think you will integrate the circumference of a cross-section of the torus around the ring.  The surface element will be:

circumference * thickness

circumference = 2pi r = 2pi (0.5) = pi
thickness = arc length = R d@  [@ is 'theta'] = 1.5 d@

Surface area =
{2pi
|    1.5 pi d@ =
}0

1.5 pi [from 0 to 2pi] =

3 pi^2 = 29.608813203268075856503472999628

You can find this calculation at:

http://www.webcalc.net/calc/0046_surfacearea.php