Hello Mike,
Basically, I’m having a little trouble understanding the explanation for #3 on the Working in Three Dimensions section on page 284. I got a little lost after the part where you explained how d was the height (this makes sense to me). Like, how do we “recognize that the ab term in the surface area expression must represent both the top and the bottom of the prism”? I think it’s somewhere from the equation but I don’t know where. I would greatly appreciate your help here Mike. Thanks!

Happy to help! If the below doesn’t get you unstuck, feel free to leave an additional question in the comments.

Once you recognize that d is the height, you know that the three vertical rectangles must all have a d term in their areas. That’s three of the four terms in the expression, and they cover ALL the surface area except the top and bottom of the prism. Right?

Put another way, you know that the whole surface area is ab + (ad + bd + cd), and you know that all the parts I just put in the parentheses make up the vertical rectangles. We haven’t accounted for the top and bottom yet, so they must both be covered in that ab term.

s-palgo says:

This explanation was great. The parentheses really helped clear up the confusion for me. Thanks Mike!