How many rectangles are there in the figure given below?

Concept

A rectangle is determined by choosing two vertical boundary lines and two horizontal boundary lines whose connecting segments are complete.

In an m-by-n rectangular grid, this is equivalent to choosing 2 of the m + 1 vertical lines and 2 of the n + 1 horizontal lines.

Application

1. The closed rectangular portion has three columns and two rows, so it has four usable vertical grid lines and three usable horizontal grid lines.

2. Choose the two vertical sides in C(4, 2) = 6 ways.

3. Choose the two horizontal sides in C(3, 2) = 3 ways.

4. Multiplying the independent choices gives 6 × 3 = 18 rectangles.

Cross-check

• By size, the grid contains 6 unit rectangles, 4 two-column single-row rectangles, 2 three-column single-row rectangles, 3 one-column double-row rectangles, 2 two-column double-row rectangles, and 1 whole-grid rectangle; these total 18.

• The pointed extension contributes no additional rectangle because its right side is a point joined by slanted lines, not a vertical segment that closes a rectangular region.

Therefore the figure contains 18 rectangles.