First, we started with this handout: (found in my Google Drive, here - Named "Sector Handout"...as always, if you want this in an editable version, just let me know!)
*Note that I made copies on separate paper...one sided and only gave this side first*
Students did this at their tables. I pushed students to include units and give both approximate and exact answers. They didn't need a "special" formula! They figured it out on their own.
After we summarized those two problems, I asked students to push themselves to come up with general formulas for both. We wrote them on the board, and several students wrote them down on that paper. Then, I asked students to do the same but in radians. They quickly figured out that the only thing that needed change in the formula was 360 becoming 2pi. (*Note here: I don't have students simplify those formulas, with the exception of S=(theta)*r. I explained to them that simplifying the formulas doesn't help my understanding of them. If they choose to do it, that's fine.)
Then, I had them put those papers away and gave them the second sheet:
I asked that they try to write (or even derive again) the formulas without looking at what we just did (I had already erased the whiteboard). They had no problem doing it! Clear to see that there was no real memorization required!
I had to help students along on problems #1 and 2, but those were awesome too. And they were able to figure them out without me standing at the board and doing examples first. Another resource for you should you want it!