Overview

Geometric counting turns diagrams into countable structures. Frequent AMC 8 patterns include counting rectangles in a grid, squares in all sizes, and lattice paths with restricted moves.

Key Ideas

• Count by size class instead of one figure at a time.
• For grids, separate counting of squares and rectangles.
• Path counting often becomes a sequence of move choices.
• Use symmetry to halve computation when the diagram is balanced.

Core Skills

Counting Rectangles in an $m\times n$ Grid

Choose 2 of the $m+1$ horizontal lines and 2 of the $n+1$ vertical lines:

Counting All Squares in an $n\times n$ Grid

Add contributions by side length:

Counting Monotone Paths

From $(0,0)$ to $(a,b)$ using only right/up moves, total paths are

Worked Example

How many rectangles are in a $2\times 3$ grid?

So the grid contains $\boxed{18}$ rectangles.

Practice Problems