Last week Grant Sanderson published an incredible video about the surface area of a sphere:

By happy coincidence my older son is spending a little time reviewing polar coordinates now. Although not exactly the same ideas, I think there’s enough overlap to make studying Sanderson’s new video worthwhile.

So, I’m going to do a 6-part project going through the video. Tonight we watched it and my son’s initial thoughts are below. Each of the next 5 parts will be spent discussing and answering the 5 questions that Sanderson asks in the 2nd half of the video.

Here’s question #1:

We’ve been away from right triangle trig a little bit lately, so I was interested to see how my son would approach this problem. His approach was a bit of a surprise, but it did get him to the right answer:

The next question in Grant’s video is about how the area of one of the rings on the sphere changes when you project it down to the “base” of the sphere (see the picture above).

I thought that answering this question would be a really good geometry, trig, and Calculus exercise for my son:

Now we get to a really interesting part -> Question #3

Grant asks you to relate the area you found in question 2 – the area of a ring around the sphere projected down to the center of the sphere – to the area of a different ring around the sphere.

Here’s my son’s work on this problem:

Finally – my son answers questions #4 and #5 after a quick review of the previous results. He was a little tired tonight, but we needed to squeeze in these two questions tonight because I have to travel for work tomorrow.

Here’s question #4:

and #5:

And here’s his work on those two questions: