Is it possible to draw a rectangle with the same area, but a smaller perimeter?

Yes.

Suppose you have a 4*1 rectangle. This would have an area of 4. The perimeter would be 4+4+1+1, which is 10.

Now suppose you have a 2*2 rectangle. This would also have an area of 4. However, this perimeter would be 2+2+2+2, which is 8. 

The second rectangle has a smaller perimeter but the same area as the first rectangle.