Yes. Start by drawing an arbitrary rectangle that hugs the state. Now rotate that rectangle, adjusting its dimensions as it turns to keep each side in contact with the shape. If you turn it through 90 degrees, then the “height” and “width” will have traded places: What had been the shorter side of the rectangle is now the longer. So at some point the two must have had the same value.

(This is a two-dimensional variation on a classic puzzle.)