What Are Similar Figures?
Similar figures are figures that have the same shape but not necessarily the same size. Formally, two figures are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. We write "figure $P$ is similar to figure $Q$" as $P \sim Q$, using the tilde symbol $\sim$.
Two photographs of the same person, a stamp-sized print and a passport-sized print, are similar: identical shape, different scale. All circles are similar to one another, and all squares are similar to one another, because their shape is fixed and only the size varies.
The Map That Lies About Size and Tells the Truth About Shape
A road map shrinks a whole country onto a page, yet every road keeps its exact shape and every junction its exact angle. That is the quiet magic of similarity: change the size as much as you like, and as long as the shape and angles survive, the two figures are "the same" in the way that matters for scale drawings, models, and photographs. Understanding when two figures are genuinely similar, and how their measurements relate, is what this article is about.
What Are the Properties of Similar Figures?
Two properties define similarity, and one more follows from them.
Corresponding angles are equal. Every matching angle in the two figures has the same measure.
Corresponding sides are proportional. Divide any side of one figure by the matching side of the other and you always get the same number, the scale factor.
Areas scale by the square of the ratio. If the sides are in ratio $\frac{x}{y}$, the areas are in ratio $\frac{x^2}{y^2}$. This one catches people out, and it has its own section below.
The scale factor is the single number that ties the two figures together: multiply every side of the smaller figure by it to get the larger.
How Do You Prove Two Triangles Are Similar?
Triangles are the workhorse of similarity, and three criteria settle whether two of them are similar triangles. You never need all six measurements, the right three are enough.
AA (Angle-Angle): if two angles of one triangle equal two angles of another, the triangles are similar. (The third pair is then automatically equal, since angles sum to $180^\circ$.)
SSS (Side-Side-Side): if all three pairs of corresponding sides are in the same ratio, the triangles are similar.
SAS (Side-Angle-Side): if two pairs of sides are in the same ratio and the included angles are equal, the triangles are similar.
Note that SSA is not a valid similarity criterion, two sides and a non-included angle do not fix a shape. The same three-angle logic is why all 30-60-90 triangles are similar to each other, and it grows directly out of the basic proportionality theorem.
How Is Similarity Different From Congruence?
Similar and congruent figures are close cousins, and mixing them up is common. Congruent figures are identical in both shape and size; similar figures share only shape.
Feature | Similar figures | Congruent figures |
|---|---|---|
Shape | Same | Same |
Size | May differ | Identical |
Corresponding angles | Equal | Equal |
Corresponding sides | Proportional (scale factor) | Equal (scale factor $= 1$) |
Symbol | $\sim$ | $\cong$ |
The clean way to hold it: all congruent figures are similar, but not all similar figures are congruent. Congruence is just similarity with a scale factor of exactly $1$.
Examples of Similar Figures
The examples build from a plain scale-up to an area trap and a real photo enlargement.
Example 1
Two rectangles are similar. The smaller is $4$ cm by $6$ cm; the larger has width $6$ cm. Find its length.
Set corresponding sides in proportion (width to width, length to length).
$$\frac{4}{6} = \frac{6}{\text{length}}$$
$$\text{length} = \frac{6 \times 6}{4} = 9 \text{ cm}$$
Final answer: the larger rectangle is $6$ cm by $9$ cm.
Example 2
Two similar figures have a scale factor of $3$. The smaller figure has an area of $5$ cm². Find the area of the larger.
The intuitive move is to multiply the area by $3$, giving $15$ cm². Test that against a picture. Scaling a shape by $3$ makes it $3$ times as wide and $3$ times as tall, so it covers far more than $3$ times the area, think of a $1 \times 1$ square becoming a $3 \times 3$ square, which holds nine unit squares, not three. Multiplying area by the linear factor undercounts badly.
Areas scale by the square of the ratio.
$$\text{Area ratio} = 3^2 = 9$$
$$\text{Larger area} = 5 \times 9 = 45 \text{ cm}^2$$
Final answer: $45$ cm², nine times the smaller area, not three.
Example 3
Triangle $ABC \sim$ triangle $DEF$. Given $AB = 4$, $DE = 6$, and $BC = 5$, find $EF$.
Corresponding sides are proportional; $AB$ matches $DE$, and $BC$ matches $EF$.
$$\frac{AB}{DE} = \frac{BC}{EF}$$
$$\frac{4}{6} = \frac{5}{EF}$$
$$EF = \frac{5 \times 6}{4} = 7.5$$
Final answer: $EF = 7.5$.
Example 4
In triangle $PQR$, $\angle P = 50^\circ$ and $\angle Q = 60^\circ$. In triangle $XYZ$, $\angle X = 50^\circ$ and $\angle Y = 60^\circ$. Are the triangles similar?
Two pairs of angles match, so by the AA criterion the triangles are similar. The third angle is fixed too: $180^\circ - 50^\circ - 60^\circ = 70^\circ$ in both.
Final answer: yes, $\triangle PQR \sim \triangle XYZ$ by AA.
Example 5
Two similar polygons have a scale factor of $\frac{5}{2}$. The smaller has a perimeter of $12$ cm. Find the larger perimeter.
Perimeter, being a length, scales by the linear factor (not its square).
$$\text{Larger perimeter} = 12 \times \frac{5}{2} = 30 \text{ cm}$$
Final answer: $30$ cm.
Example 6
A photograph is $6$ inches wide and $4$ inches tall. It is enlarged similarly to a width of $15$ inches. Find the new height.
Similar enlargement keeps the shape, so both dimensions share one scale factor.
$$\text{Scale factor} = \frac{15}{6} = 2.5$$
$$\text{New height} = 4 \times 2.5 = 10 \text{ inches}$$
Final answer: the enlarged photo is $10$ inches tall. (Enlarge to a different height, say $8$ inches, and the shape would distort, no longer similar.)
Why Does Scaling Shape Change More Than Size?
Similarity is the mathematics of scale models, maps, and blueprints, but it comes with a warning that engineers learned the hard way. When you scale a figure up, its lengths grow by the scale factor, its areas grow by the square of that factor, and its volumes grow by the cube. This is the square-cube law, and it is why you cannot simply build a big thing by scaling up a small one.
Scale models mislead: a model bridge that holds its own weight beautifully tells you little about the real bridge, because weight (volume) outruns supporting cross-section (area) as size grows. The Wikipedia entry on the square-cube law traces this back to Galileo.
Maps and drawings: a $1 : 50000$ map is a figure similar to the land, the same shape, a fixed scale relating map distance to real distance.
Why areas need care: because the area of similar figures scales by the square of the side ratio, doubling a shape quadruples the paint, tile, or fabric it needs.
For a formal treatment of the similarity relation, the Wolfram MathWorld entry on similarity is a solid authoritative reference. Similarity itself is built on ordinary triangles and proportion.
What Are the Most Common Mistakes With Similar Figures?
Mistake 1: Scaling area by the linear factor
Where it slips in: any problem that jumps from side ratios to area or volume.
Don't do this: multiply area by the scale factor. That is the single most reliable error here, treating a two-dimensional quantity as if it grew in one dimension.
The correct way: square the scale factor for area, cube it for volume. Scale factor $3$ means area $\times 9$ and volume $\times 27$.
Mistake 2: Matching the wrong corresponding sides
Where it slips in: when the two figures are drawn in different orientations, so the sides do not line up on the page.
Don't do this: pair sides by position on the paper.
The correct way: match sides by the angles they sit between, using the similarity statement's letter order. In $\triangle ABC \sim \triangle DEF$, side $AB$ always corresponds to $DE$, whatever way the triangles are turned.
Mistake 3: Assuming any two figures of the same type are similar
Where it slips in: with rectangles, triangles, and other polygons that are not regular.
Don't do this: treat all rectangles as similar. A $2 \times 3$ rectangle and a $2 \times 5$ rectangle have equal angles but non-proportional sides, so they are not similar.
The correct way: check both conditions, equal angles and proportional sides. (All squares and all circles pass automatically; most other shapes do not.)
Conclusion
Similar figures have equal corresponding angles and proportional corresponding sides, written with the symbol $\sim$.
The constant ratio between corresponding sides is the scale factor; triangles are proven similar by AA, SSS, or SAS.
Areas scale by the square of the ratio and volumes by the cube, the square-cube law.
All congruent figures are similar, but not all similar figures are congruent.
To build similarity and scale reasoning with a teacher, explore Bhanzu's geometry tutor or high school math tutor, or join a live math class online.
A Practical Next Step
Work through the exercises above, then try this: if two similar solids have a scale factor of $2$, what happens to their surface area and their volume? If you are unsure, return to the square-cube reasoning in Mistake 1. Want a live trainer to check your corresponding-side matching? Book a free demo class.
Read More
Congruence in triangles, the same-size, same-shape case of the similarity idea.
Proportion, the equal-ratio foundation under every similarity statement.
Types of triangles, the triangles you most often test for similarity.
Triangle congruence theorem, how congruence criteria mirror the similarity ones.
Quadrilaterals, four-sided figures and when two of them are similar.
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