Hey there!
Area- defines the amount of space inside a boundary
Perimeter- defines the distance around this boundary
Considering we're looking for fencing that goes around the backyard, not in it, it seems reasonable that we would use perimeter. Remember, we're not covering the entire backyard with fencing- in which case we would need to know the area. In this case, we're just looking at the distance around the backyard.
Richard would probably need the distance in whatever units he's using- meters, yards, feet- depending on how big his yard is. A common misconception here is that you would use square feet. Remember that square feet is for area, as you write units squared after you do any area calculation.
Hope this helps!
