What Are Similar Triangles?
Similar triangles are triangles that have exactly the same shape but possibly different sizes. Two triangles are similar when both of these hold at once:
Their corresponding angles are equal, and
Their corresponding sides are in the same ratio (proportional).
One triangle is then a scaled copy of the other, enlarged or shrunk by a constant factor called the scale factor, and it stays similar even if it is rotated or flipped into a mirror image. We write similarity with the symbol $\sim$: $\triangle ABC \sim \triangle DEF$ means triangle $ABC$ is similar to triangle $DEF$, with the vertices listed in matching order so you know which angle pairs with which.
What Are the Similarity Criteria (AA, SAS, SSS)?
A common reader question is whether you really have to check all the angles and all the sides. You do not. Three shortcut tests, the similarity criteria, each confirm similarity from partial information.
AA (Angle-Angle). If two angles of one triangle equal two angles of another, the triangles are similar. Why two is enough: the three angles of any triangle add to 180°, so once two angles match, the third must match automatically. (This is sometimes written AAA, but checking two is all you need.)
SAS (Side-Angle-Side). If two pairs of sides are in the same ratio and the angles between them are equal, the triangles are similar. The equal angle must be the one included between the two proportional sides.
SSS (Side-Side-Side). If all three pairs of corresponding sides are in the same ratio, the triangles are similar. No angle check is needed, because three matched side ratios lock the shape completely.
Properties of Similar Triangles
Once two triangles are known to be similar, a useful list of facts comes free:
All corresponding angles are equal. Shape is preserved exactly.
All corresponding sides are in the same ratio. That common ratio is the scale factor $k$.
The ratio of perimeters equals the scale factor. Perimeter scales the same way the sides do.
The ratio of areas equals the square of the scale factor. If the sides are in ratio $k$, the areas are in ratio $k^2$ — covered on its own below, because it trips up the most students.
Corresponding medians, altitudes, and angle bisectors are all in the ratio $k$. Every matching length scales by the same factor as the sides.
The Area Ratio Rule
This property deserves its own heading because it is the one students most often get wrong. If two triangles are similar with sides in the ratio $k$, then their areas are in the ratio $k^2$, the square of the side ratio.
The reason is that area depends on two dimensions, base and height, and both scale by $k$. Multiplying two lengths that have each grown by $k$ multiplies the area by $k \times k = k^2$:
$$\frac{\text{Area}_1}{\text{Area}_2} = \left( \frac{\text{side}_1}{\text{side}_2} \right)^2 = k^2.$$
So a triangle with sides twice as long does not have twice the area, it has four times the area. A scale factor of 3 means nine times the area. This squaring is exactly why a pizza of double the diameter feeds far more than two people.
Similar Triangles vs Congruent Triangles
Similar and congruent are the two ways triangles can "match", and the difference is just one word: size. Similar triangles have the same shape, possibly different sizes. Congruent triangles have the same shape and the same size — they are similar triangles with a scale factor of exactly 1.
Feature | Similar ($\sim$) | Congruent ($\cong$) |
|---|---|---|
Shape | Same | Same |
Size | May differ | Identical |
Corresponding angles | Equal | Equal |
Corresponding sides | Proportional | Equal (ratio 1:1) |
Scale factor | Any positive number | Exactly 1 |
Test examples | AA, SAS, SSS | SSS, SAS, ASA, RHS |
Every congruent pair is also similar (with $k = 1$), but most similar pairs are not congruent. (For the equal-size case, see our sibling article on congruent triangles.)
Examples of Similar Triangles
With the criteria, the properties, and the area rule in place, here is similarity doing real work. The problems build from a direct proportion up to area ratios and the shadow problem from the hook.
Example 1 - $\triangle ABC \sim \triangle DEF$ with $AB = 4$, $DE = 6$, and $BC = 5$. Find $EF$.
Corresponding sides are in the same ratio, and $AB$ pairs with $DE$, $BC$ with $EF$:
$$\frac{AB}{DE} = \frac{BC}{EF} ;\Rightarrow; \frac{4}{6} = \frac{5}{EF} ;\Rightarrow; EF = \frac{5 \times 6}{4} = 7.5.$$
Final answer: $EF = 7.5$.
Example 2 - Two similar triangles have sides in the ratio 2:3. The smaller has an area of 16 cm². Find the area of the larger.
A tempting first move is to scale the area by the same 2:3 ratio, giving $16 \times \tfrac{3}{2} = 24$ cm². Check the reasoning: area covers a surface, and both the base and the height grow by the ratio, so the area must grow by the ratio squared, not the ratio itself. Scaling area linearly ignores the second dimension entirely.
Done correctly, the area ratio is $(2:3)^2 = 4:9$:
$$\frac{16}{\text{Area}{\text{large}}} = \frac{4}{9} ;\Rightarrow; \text{Area}{\text{large}} = \frac{16 \times 9}{4} = 36 \text{ cm}^2.$$
Final answer: 36 cm².
Example 3 - Are two triangles similar if one has angles 40° and 75°, and the other has angles 75° and 65°?
Find the third angle of each. First: $180° - 40° - 75° = 65°$, so its angles are 40°, 75°, 65°. Second: $180° - 75° - 65° = 40°$, so its angles are 75°, 65°, 40°. Both triangles have angles 40°, 65°, 75°.
By the AA criterion, the triangles are similar.
Final answer: yes, they are similar (AA).
Example 4 - A 6-foot person casts a 4-foot shadow at the same time a tree casts a 30-foot shadow. How tall is the tree?
The person and the tree make similar right triangles (same sun angle), so their height-to-shadow ratios are equal:
$$\frac{\text{height}}{\text{shadow}}: \quad \frac{6}{4} = \frac{H}{30} ;\Rightarrow; H = \frac{6 \times 30}{4} = 45 \text{ feet}.$$
Final answer: the tree is 45 feet tall.
Example 5 - In $\triangle ABC$, a line $DE$ is drawn parallel to side $BC$, meeting $AB$ at $D$ and $AC$ at $E$. If $AD = 3$, $DB = 6$, and $AE = 4$, find $EC$.
Because $DE \parallel BC$, triangle $ADE$ is similar to triangle $ABC$, and the Basic Proportionality Theorem (Thales's theorem) gives $\tfrac{AD}{DB} = \tfrac{AE}{EC}$:
$$\frac{3}{6} = \frac{4}{EC} ;\Rightarrow; EC = \frac{4 \times 6}{3} = 8.$$
Final answer: $EC = 8$.
Example 6 - Two similar triangles have corresponding sides 8 cm and 12 cm. Find the ratio of their perimeters and the ratio of their areas.
The scale factor is $\tfrac{8}{12} = \tfrac{2}{3}$. Perimeters scale by the same factor, areas by its square:
$$\text{perimeter ratio} = \frac{2}{3}, \qquad \text{area ratio} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}.$$
Final answer: perimeters 2:3; areas 4:9.
Why Similar Triangles Matter
Similarity is the geometry of scaling, and scaling is everywhere humans build, measure, or picture the world.
Measuring the unreachable. The shadow method (Example 4) is how the ancient Greek Thales is said to have measured the height of the Great Pyramid, and how surveyors still estimate heights and distances they cannot reach directly.
Maps and scale models. Every map, blueprint, and architectural model is a similar copy of the real thing: the scale on a map (say 1:50,000) is exactly a similarity ratio, and the area-squared rule is why a region four times wider on the map covers sixteen times the ground.
Cameras and screens. A camera lens projects a similar, scaled-down triangle of light onto the sensor; the same geometry decides how large an object appears at a given distance, which is the foundation of perspective in art and 3D graphics.
Trigonometry's foundation. The sine, cosine, and tangent ratios only make sense because all right triangles with a given acute angle are similar, so their side ratios are fixed regardless of size — the fact that makes trigonometry possible at all.
For a Grade 10 student, similar triangles are the gateway to trigonometry and coordinate geometry both, master the ratios now and two later chapters arrive already half-built.
Where Students Trip Up on Similar Triangles
Mistake 1: Scaling area by the side ratio instead of its square
Where it slips in: Given the side ratio and one area, the student multiplies the area by the side ratio directly.
Don't do this: Use $\text{Area}_2 = \text{Area}_1 \times k$ for a side ratio $k$.
The correct way: Area scales by $k^2$, not $k$, because both base and height stretch by $k$. A side ratio of 3 means an area ratio of 9. Picture the whole surface growing in two directions, not one.
Mistake 2: Pairing the wrong corresponding sides
Where it slips in: The student matches sides by their position on the page rather than by the vertex order in $\triangle ABC \sim \triangle DEF$.
Don't do this: Set up a proportion with sides that are not actually corresponding.
The correct way: The similarity statement tells you the pairing: in $\triangle ABC \sim \triangle DEF$, $A$ pairs with $D$, $B$ with $E$, $C$ with $F$, so side $AB$ corresponds to $DE$, $BC$ to $EF$, $CA$ to $FD$. The rusher who reads off sides by position often builds a correct-looking proportion that pairs the wrong sides.
Mistake 3: Confusing similar with congruent
Where it slips in: The student concludes the triangles are equal in size, or insists all three sides must be equal for similarity.
Don't do this: Treat similar triangles as identical, or demand equal side lengths.
The correct way: Similar means same shape, possibly different size — equal angles and proportional sides. Congruent is the special case where the scale factor is exactly 1. The second-guesser who knows both terms still has to check whether the problem asks for shape (similar) or shape-and-size (congruent).
Key Takeaways
Similar triangles have the same shape but possibly different sizes: equal corresponding angles and proportional corresponding sides.
Similarity is proved by any one of three criteria — AA, SAS, or SSS.
Corresponding sides, perimeters, medians, and altitudes all scale by the same factor $k$; areas scale by $k^2$.
Congruent triangles are the special case of similar triangles with a scale factor of exactly 1.
The most common error is scaling area by the side ratio instead of its square; remember area grows in two dimensions.
Practice These Problems to Solidify Your Understanding
$\triangle PQR \sim \triangle XYZ$ with $PQ = 9$, $XY = 12$, and $QR = 15$. Find $YZ$.
Two similar triangles have sides in the ratio 5:2. The larger has an area of 100 cm². Find the area of the smaller.
A 5-foot pole casts a 3-foot shadow while a building casts a 24-foot shadow at the same time. Find the building's height.
Answer to Question 1: $YZ = 20$. Answer to Question 2: $\left(\tfrac{2}{5}\right)^2 \times 100 = \tfrac{4}{25} \times 100 = 16$ cm². Answer to Question 3: $\tfrac{5}{3} = \tfrac{H}{24}$, so $H = 40$ feet. If Question 2 gave 40 cm², check that you squared the ratio before scaling the area (see Mistake 1).
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