How many rectangles are there in the given figure?

To count all rectangles in a grid-based figure, work by size, from smallest to largest: first count the unit (smallest) rectangles bounded by adjacent grid lines, then the rectangles formed by joining two adjacent units, then four units, and so on, up to the single largest rectangle that encloses the whole grid. A region only counts as a rectangle if it is a closed, four-sided figure with all right angles; a triangular region attached to the grid is not a rectangle, however it looks.

Label the figure as shown. The rectangular grid runs from A (top-left) to C (top-right) to E (bottom-right) to G (bottom-left), with B and F marking the midpoints of the top and bottom edges, and I and K marking the midpoints of the left and right edges of this rectangle; J is the centre point where the two mid-lines cross. (The pointed flaps at H and D are triangles, not rectangles, so they add nothing to the count.)

1. Unit (smallest) rectangles, one per grid cell: ABJI, BCKJ, IJFG, JKEF, 4 rectangles.

2. Two-cell rectangles, formed by joining two adjacent units: ACKI (top half), IKEG (bottom half), ABFG (left half), BCEF (right half), 4 rectangles.

3. Four-cell rectangle, the single rectangle spanning the whole grid: ACEG, 1 rectangle.

Total = 4 + 4 + 1 = 9 rectangles.

This matches the general formula for a grid of m rows and n columns of unit cells: number of rectangles = C(m+1, 2) times C(n+1, 2). Here m = n = 2, giving C(3,2) times C(3,2) = 3 times 3 = 9, confirming the count.