Longmont, CO, April 2025
While walking through the cereal aisle, John spotted math on a cereal box.
He took a picture because this challenge could not be ignored. As it turns out, this was a necessary evil. Almost everything about this marketing blurb is wrong.
Problem #1
The surface area of a torus (i.e., doughnut) on the package is just wrong. The surface area of a torus is 2πR * 2πr, where R is the radius of the doughnut and r is the radius of the circle that makes the cross section of the doughnut. The packaging is off by 2x. It should be 4π2Rr.
Problem #2
Without defining the ratio of R to r, their equations prove nothing. Their assertion is that:
| Sphere Surface Area | > | Torus Surface Area |
| 4πR2 | > | 4π2Rr |
| (4πR) R | > | (4πR) πr |
| R | > | πr |
| R/π | > | r |
| .318 R | > | r |
If πr > R, their claims of more frosting are completely false. All this requires is that r be more than about one third of R.
Problem #3
The box implies that R is the same for the sphere and torus. The best assumption that can be made is that the surface area calculations should be for equal volumes of cereal.
| Sphere Volume | = | Torus Volume |
| (4πR3)/3 | = | (πr2) (2πR) |
| (2πR) (2R2/3) | = | (πr2) (2πR) |
| 2R2/3 | = | πr2 |
| (2/3π) R2 | = | r2 |
| .461 R | = | r |
We can plug the value of r for equal volume into the inequality from problem #2.
| Sphere Surface Area | > | Torus Surface Area |
| .318 R | > | r |
| .318 R | > | .461 R |
| . |
Doh! Last time I checked, .318 was not greater than .416. We should not expect an equal volume of spherical shaped cereal to have more frosting than torus shaped cereal, assuming they can successfully frost the entire surface area of both shapes.
Conclusion
A sphere has the smallest surface area to volume of any three dimensional shape. This may not be something that the average person knows, but you would think that the cereal manufacturer could hire a math expert for an hour to verify this. Paying their marketing department for the 30 minutes of Google searching required to debunk this might have worked too. Either way,
- They did the math wrong
- This is arbitrary marketing nonsense
- Or, they did the math right to give you the least frosting and lied about it