Perfect Rectangle — Definition, Formula & Examples

A perfect rectangle (or squared rectangle) is a rectangle that can be cut into smaller squares where every square has a different side length. No two squares in the tiling share the same size.

A perfect rectangle is a rectangle that admits a tiling by finitely many squares, each of mutually distinct side lengths. Formally, given a rectangle of dimensions $a \times b$ , there exist $n$ squares with pairwise distinct positive integer side lengths $s_1, s_2, \ldots, s_n$ that partition the rectangle without gaps or overlaps.

How It Works

To verify whether a rectangle is a perfect rectangle, you check two conditions: the entire interior must be covered by non-overlapping squares, and every square must have a unique side length. The simplest known perfect rectangle uses 9 squares and has dimensions

32 \times 33

. Finding these tilings is a challenging combinatorial problem that connects geometry to graph theory and electrical network analogies. Researchers discovered that modeling the rectangle as an electrical circuit — where each square corresponds to a resistor — provides a systematic way to find valid tilings.

Worked Example

Problem: The simplest perfect rectangle has dimensions 32 × 33 and is tiled by 9 squares. The square side lengths are 1, 5, 8, 9, 10, 14, 15, 18, and 33 is the height. Verify that the total area of the 9 squares equals the area of the rectangle.

Step 1: Compute the area of the rectangle.

A_{\text{rect}} = 32 \times 33 = 1056

Step 2: The 9 square side lengths are 1, 5, 8, 9, 10, 14, 15, 18, and 4. Sum their squared areas.

1^2 + 4^2 + 5^2 + 8^2 + 9^2 + 10^2 + 14^2 + 15^2 + 18^2 = 1 + 16 + 25 + 64 + 81 + 100 + 196 + 225 + 324 = 1032

Step 3: Note: the correct set of 9 squares for the 32 × 33 rectangle has side lengths 1, 5, 8, 9, 10, 14, 15, 18, and we need to recheck. The actual simplest perfect rectangle (by fewest squares) is the 32 × 33 rectangle with squares of sides 18, 15, 14, 10, 9, 8, 7, 4, 1. Let us verify:

18^2 + 15^2 + 14^2 + 10^2 + 9^2 + 8^2 + 7^2 + 4^2 + 1^2 = 324 + 225 + 196 + 100 + 81 + 64 + 49 + 16 + 1 = 1056

Answer: The sum of the 9 square areas equals

1056 = 32 \times 33

, confirming the tiling is valid. The side lengths 18, 15, 14, 10, 9, 8, 7, 4, 1 are all distinct, so this is a perfect rectangle.

Why It Matters

Perfect rectangles appear in recreational mathematics and combinatorics competitions. The electrical network analogy used to discover them is taught in discrete mathematics and graph theory courses, connecting geometry to circuit analysis in a surprising way.

Common Mistakes

Mistake: Confusing a perfect rectangle with a perfect square (squared square).

Correction: A perfect rectangle tiles a non-square rectangle with distinct squares. A squared square tiles a square with distinct squares — a harder problem first solved in 1939 (order 69) and later with fewer squares.