Embedded Figures & Counting of Figures Notes

Embedded Figures & Counting of Figures

Finding a given shape hidden inside a larger drawing, and the three checks that settle it — same size, same proportions, no rotation unless allowed; counting triangles, squares and rectangles in a figure without missing any or counting twice; the labelling method and the formulas for a grid; counting rectangles in an m × n grid; counting triangles in a triangle divided by lines; straight lines and parallelograms in standard figures — with fully worked examples.

Last reviewed 22 Sept 2026 · 4 min read

Embedded figures

You are given a small shape and a complex drawing, and asked which drawing contains the shape exactly as it is — same size and same proportions, and usually in the same orientation.

Three checks answer nearly every question:

Size: the embedded copy must be the same size, not a scaled one.
Proportions: every angle and relative length must match.
Orientation: unless the question allows rotation, the shape must appear the right way up.

Exam TipUse one distinctive part

Pick the most unusual feature of the small shape — a notch, a slanted line, a corner cut — and look only for that in each option. It eliminates three options far faster than tracing the whole outline.

Counting figures without missing any

The only reliable method is to label and be systematic:

Label every vertex and intersection with a letter.
Count the simplest figures first — the smallest triangles or squares.
Then count the figures made of two of those, then three, and so on.
Add the totals, and sanity-check against a formula where one exists.

Counting "as you see them" is what produces answers that are two or three short.

Formulas for grids

FormulaSquares and rectangles in a grid

In a grid of $m$ columns and $n$ rows of unit squares:

Rectangles = \frac{m ( m + 1 )}{2} \times \frac{n ( n + 1 )}{2}

Squares (when m = n) = 1^{2} + 2^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

For a chessboard ( $8 \times 8$ ): $1296$ rectangles and $204$ squares.

A rectangle is chosen by picking two of the $m + 1$ vertical lines and two of the $n + 1$ horizontal ones, which is where the formula comes from.

Triangles in a divided triangle

A triangle cut by lines from one vertex to the opposite side, into $n$ small triangles, contains

\frac{n ( n + 1 )}{2}

triangles in all. If the triangle is also divided by a line parallel to the base, count the new small triangles and the larger ones they combine into, layer by layer.

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Embedded Figures & Counting of Figures

Embedded figures

Counting figures without missing any

Formulas for grids

Triangles in a divided triangle

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