Question:
A solid sphere is cut into $16$ identical pieces with $5$ cuts.
Find the percentage increase in the new total surface area of all the pieces over that of the original sphere?
My try:
Initially I was confused about how to make such cuts on a sphere!
But then I was told that the cuts would be like an asterixis but with $8$ equally spaced spikes on the cross-section. The $5^{th}$ cut would be parallel to the table, thus dividing the sphere in $16$ identical pieces.
After this, for a piece, I was able to imagine and get the surface area of $2$ ($1$ curved and $1$ flat) out of the $4$ surfaces. I was unable to picturize mentally the rest of the $2$ (identical) faces.
The areas of those $2$ surfaces, $1$ curved and $1$ flat, would be $\frac1{16}4\pi r^2$ and $\frac18\pi r^2$ respectively.
I know the answer is $250\%$ but that wasn't really useful. I did trace back from there to find the area of the remaining $2$ surfaces but ended up using a lot of time though I should be using much less.
How to go about solving this question? Am I using the right approach?
Best Answer
Every cut gives an additional surface equivalent with 2 times a full circle with radius $r$ (one circle at each side of the cut), imagine this when splitting a sphere exactly in half. So with 5 cuts you have in total 10 circle surfaces extra is $10\pi r^2$. The sphere was $4\pi r^2$, so the new surfaces are 250% of what you had.