Similar figures

Hold a photograph next to its enlarged poster print and something stays fixed even though the size has clearly changed: the shape. Every side of the poster is the same number of times longer than the matching side of the photograph — not roughly, but exactly, for every single pair of sides at once. That single fixed multiplier is what mathematics calls a scale factor, and any two figures connected by one are called similar figures. Similar figures sit inside the Proportional Reasoning chapter of Class 8 because they are, underneath the geometry, just another direct-proportion problem: know three of the four related lengths and the fourth is forced. This section builds the idea from "same shape, different size" up to a clean formula, one worked poster-and-flag problem, and the traps that catch students who match up the wrong sides.

The core insight

Two figures are similar when their corresponding angles are equal and every pair of corresponding sides shares one fixed ratio, the scale factor $k$ , so any unknown side can be found by solving $\dfrac{a}{b}=k$ .

The core idea

Two figures are similar if one is an enlargement or a reduction of the other. That single idea splits into two testable conditions:

Corresponding angles are equal. The shape's "turning" at each vertex is preserved — a similar figure is never stretched unevenly, only scaled uniformly.
Corresponding sides are in the same ratio. If a side doubles, every side doubles; nothing scales by a different amount.

For a figure with sides $a_1, a_2, a_3, \dots$ similar to a figure with corresponding sides $b_1, b_2, b_3, \dots$ , both conditions are captured in one equation:

\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} = \dots = k

where: $a_1, a_2, \dots$ are the sides of the first figure, $b_1, b_2, \dots$ are the corresponding sides of the second figure (each $b_i$ paired with the $a_i$ that plays the same role in the shape), and $k$ is the constant scale factor connecting figure $A$ to figure $B$ . The value of $k$ tells you what kind of similarity you're looking at: $k>1$ means the second figure is an enlargement, $k<1$ means it is a reduction, and $k=1$ means the two figures are congruent — identical in size and shape, which is really just similarity with no scaling at all. Because every linear measurement obeys the same $k$ , an unknown side is always recoverable from three known ones by cross-multiplying the equal-ratio equation, exactly the way any direct-proportion question is solved.

Figure

A small rectangle labelled photo ten by fifteen centimetres next to a larger similar rectangle labelled poster twenty five by thirty seven point five centimetres with matching sides joined by dashed lines marked scale factor two point five

Derivation ACorresponding sides give a scale factor

The goal of this derivation is a one-line rule that is longer to state carefully than it is to use afterward: if two figures are similar, then picking any single matched pair of sides and dividing gives the same number $k$ that you would get from any other matched pair, and that shared $k$ can then be applied directly to find every missing length in the figure without re-deriving it each time from scratch.

Step 1. Write down what similarity guarantees. By definition, for every pair of corresponding sides, $\dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} = \dots = k$ . This is not an assumption to check side-by-side each time — it is the definition of similar figures, so once you already know the two figures are similar, the ratio equality is guaranteed true by construction, for every pair, not just one.

Step 2. Pick any one matched pair whose two lengths are both known, and compute $k$ directly by dividing the second figure's side by the first figure's corresponding side (or the reverse, as long as the direction of division stays consistent for every later step).

Step 3. Apply that same $k$ to any other side of the first figure to get the matching side of the second figure: multiply the known side of figure $A$ by $k$ to land on the unknown side of figure $B$ , or equivalently solve the cross-multiplied equation $a \times (\text{unknown}) = b \times (\text{known pair})$ when the unknown sits on the other figure instead.

Step 4. Sanity-check by computing $k$ a second way from a different pair of sides, whenever the problem supplies enough data to do so. Since $k$ must be exactly the same number for every corresponding pair, a mismatch between the two computed values is a reliable signal that the wrong sides were paired as "corresponding" in Step 2.

Worked example

Problem. A photograph is 10 cm wide and 15 cm long. It is enlarged into a poster that is 25 cm wide, keeping the same shape. Find the length of the poster.

Step 1. Since the poster is similar to the photograph, set up the equal-ratio equation using width and length as the two known-and-related sides:

\frac{10}{15} = \frac{25}{x}

Step 2. Cross-multiply: $10 \times x = 15 \times 25 = 375$ .

Step 3. Solve: $x = \dfrac{375}{10} = 37.5$ .

Answer: the poster is 37.5 cm long.

Check the scale factor. $k = \dfrac{25}{10} = 2.5$ ; applying it to the length directly, $15 \times 2.5 = 37.5\ \text{cm}$ — matches.

Competitive stretch

Two rectangular flags are similar. The smaller flag is 18 cm by 12 cm; the larger flag's shorter side is 30 cm. Step 1. Match corresponding sides: the smaller flag's 12 cm shorter side pairs with the larger flag's 30 cm shorter side, so $k = \dfrac{30}{12} = 2.5$ . Step 2. Apply $k$ to the smaller flag's 18 cm longer side: $x = 18 \times 2.5 = 45$ . The larger flag's longer side is 45 cm, and the same $k=2.5$ used in the poster example reappears here — a reminder that the scale factor is a property of the pair of figures, not of any one side.

Common traps

Pairing the wrong sides. Only sides that play the same role in each figure (both shorter sides, or both longer sides) are "corresponding." Matching a short side of one figure to a long side of the other gives a scale factor that means nothing.
Assuming same shape category means similar. A 4 cm by 6 cm rectangle and a 5 cm by 6 cm rectangle are both rectangles but are not similar, because $4/5 \neq 6/6$ .
Confusing similar with congruent. Similar figures need only the same shape ( $k$ can be anything); congruent figures need $k=1$ exactly. Congruent figures are always similar, but similar figures are not always congruent.
Scaling only the side asked about. The same $k$ applies to every linear measurement in the figure — width, length, diagonals, perimeter — not just the one side the question happens to mention.

Ask the tutor: stuck on which sides correspond in a similarity problem — sketch your figure and ask the AI tutor to walk through the matching with you.

How do I know which sides correspond in two similar figures?

Look for sides that play the same role in the shape, not sides that simply look close in the diagram. In similar rectangles, match shorter side to shorter side and longer side to longer side; in more general similar figures, corresponding sides are the ones between vertices that map onto each other when one figure is scaled onto the other. If a problem gives you a sketch, matching angles (equal in both figures) is the safest way to identify which sides correspond before writing any ratio.

Is every rectangle similar to every other rectangle?

No. Two rectangles are similar only if the ratio of their two sides matches. A 4 cm by 6 cm rectangle has side ratio $4:6 = 2:3$ ; a 5 cm by 6 cm rectangle has ratio $5:6$ , which is different. Since $2:3 \neq 5:6$ , these two rectangles are not similar even though both are rectangles. Similarity depends on the actual proportion between sides, not on the general shape category.

What is the difference between similar and congruent figures?

Similar figures have the same shape — equal corresponding angles and proportional corresponding sides — but can be different sizes, with any scale factor $k$ . Congruent figures are a special case where $k=1$ exactly, meaning the two figures are identical in both shape and size, so every corresponding side is equal in length, not just proportional. Every congruent pair is automatically similar, but most similar pairs are not congruent.

Similar figures

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads