My young son was asked to derive the surface area of a sphere using pure algebra. He could not get to the right formula but it seems that his reasoning is right. Please tell me what's wrong with his logic.
He reasons as follows:
- 1.Slice a sphere into thin circles (or rings if you hollow them out)
- The sum of the circumferences of all the circles forms the surface area of the sphere.
- Since the formula of circumference is $2{\pi}R$, the sum of the circumferences would be $2{\pi}(R_1+R_2+R_3+…+R_n)$
- My son draws the radii beside each other and concludes that their sum would be equivalent to one-half of the area of the largest circle
or $({\pi}R^2)/2$. He appears to be right from his drawing. - Substituting the sum of the radii, he comes up with a formula for the surface area of the sphere as $(R^2)*(\pi^2)$.
- He asks me what's wrong with his procedure that he cannot derive the correct formula of $4{\pi}R^2$.
Best Answer
In this answer, it is shown geometrically that the area of a strip of sphere between two parallels of latitude is the same as the area of the orthogonal projection of that strip onto the cylinder with the same radius as the sphere and whose axis is parallel to the north-south axis of the sphere.
Finding the area of this cylinder is just multiplying its height, $2r$, by its circumference, $2\pi r$. Thus, the area of the sphere is $4\pi r^2$.
Your son's procedure seems to be summing the circumferences of the green strips on the sphere, but there must be some accounting for the widths of these strips. This seems to be the problem he is encountering.