Squares by Area

The area of a square is its side squared. A square lined up with the grid can only have area 1, 4, 9, 16 and so on. A tilted square can have other areas: a side with the step ⟨a, b⟩ has length √(a² + b²), so the square has area a² + b².

Here each side goes 2 across and 1 up, so it is √5 long and the area is 5. To draw it, take the step ⟨2, 1⟩, then turn it a quarter for the next side, ⟨-1, 2⟩, then ⟨-2, -1⟩, then ⟨1, -2⟩ back to the start. The quarter turns make every corner a right angle, and every side has the same length.