Gnomon — Definition, Formula & Examples

A gnomon is the L-shaped figure that remains when a smaller parallelogram is removed from a corner of a larger similar parallelogram. The term originates from ancient Greek geometry and describes any shape that, when added to a given figure, produces a larger figure similar to the original.

Given a parallelogram $P$ and a smaller similar parallelogram $P'$ placed at one vertex so that corresponding sides are parallel and share that vertex, the gnomon is the region $P \setminus P'$ . More generally, a gnomon to a polygon $S$ is a figure $G$ such that $S \cup G$ is similar to $S$ .

How It Works

Start with a parallelogram and place a smaller, similar parallelogram in one corner so their sides align. The remaining L-shaped region is the gnomon. This idea extends beyond parallelograms: in number theory, the ancient Greeks used gnomons with square numbers, noting that adding consecutive odd numbers (arranged as an L-shape of dots) to a square array produces the next perfect square. The concept captures the idea of growth that preserves shape.

Worked Example

Problem: A square has side length 4. A smaller square of side length 2 is placed in its top-right corner. Find the area of the resulting gnomon.

Step 1: Find the area of the larger square.

A_{\text{large}} = 4^2 = 16

Step 2: Find the area of the smaller square removed from the corner.

A_{\text{small}} = 2^2 = 4

Step 3: Subtract to find the area of the L-shaped gnomon.

A_{\text{gnomon}} = 16 - 4 = 12

Answer: The gnomon has an area of 12 square units.

Why It Matters

Gnomons appear in Euclid's Elements (Book II) and are foundational to classical proofs about areas and proportions. Understanding gnomons helps in studying figurate numbers, where adding a gnomon of dots to a triangular or square arrangement produces the next figurate number in the sequence. The concept also connects to the golden ratio, since a gnomon added to a golden rectangle produces a larger golden rectangle.

Common Mistakes

Mistake: Assuming a gnomon must be L-shaped from squares only.

Correction: A gnomon applies to any parallelogram (or more generally, any polygon) where removing a similar copy from a corner leaves the remaining region. The shape of the gnomon depends on the original figure.