Diagonal units

Akshi and Toshi look at a triangle drawn on a dot grid. The dots are in rows and columns, like the corners of little squares. Two sides of the triangle go straight along the grid lines, each 3 units long, and they meet at a square corner. The third side goes slantwise, from corner to corner across 3 little squares.

Akshi counts the pieces between dots: 3 + 3 + 3 = 9, so she says the perimeter is 9 units. Toshi says it must be more than 9 units. Who is right?

Toshi is right. The book gives the two kinds of step their own names:

• A step along a grid line is a straight unit (s).
• A step from one corner of a little square to the opposite corner is a diagonal unit (d).

The diagonal of a square is longer than its side (cut a square paper corner to corner and lay the cut edge beside a side to see it), so d is longer than s. The triangle has 6 straight units and 3 diagonal units, and we write its perimeter as 6s + 3d units. Because each d is more than 1 unit, 6s + 3d is more than 9 units.

So on a dot grid, count the straight steps and the diagonal steps separately, and never add them together as if they were the same length. A shape with 4 straight steps and 2 diagonal steps has perimeter 4s + 2d units, not 6 units.