SSS and SAS Similarity

Prove triangles similar from side ratios (SSS) or two side ratios and the included angle (SAS), then use the scale factor for sides, perimeter and area.

Advanced30 min18 problems
Practice Problems

Definition

In AA Similarity you proved triangles similar with angles only. Now you will do it with sides. Remember: similar triangles have the same shape, their corresponding sides are in the same ratio, the scale factor kk.
SSS Similarity (Side-Side-Side): if all three pairs of corresponding sides are in the same ratio, the triangles are similar.
DEAB=EFBC=FDCA=k⇒△ABC∼△DEF\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k \quad\Rightarrow\quad \triangle ABC \sim \triangle DEF
SAS Similarity (Side-Angle-Side): if two pairs of corresponding sides are in the same ratio and the included angle (the angle between those two sides) is congruent, the triangles are similar.
DEAB=EFBC=k and ∠B=∠E⇒△ABC∼△DEF\frac{DE}{AB} = \frac{EF}{BC} = k \text{ and } \angle B = \angle E \quad\Rightarrow\quad \triangle ABC \sim \triangle DEF
How to match the sides: when you only know the lengths, sort the sides of each triangle from shortest to longest and pair them in that order: shortest with shortest, middle with middle, longest with longest.
Once you know the triangles are similar:
  • every side of the image is kk times the matching side
  • the perimeter ratio is kk
  • the area ratio is k2k^2
Compare with congruence: SSS and SAS congruence need equal sides (k=1k = 1), SSS and SAS similarity only need proportional sides.

Try it now

Which triangle is similar to a triangle with sides 22, 33 and 44?

Worked Examples

△ABC\triangle ABC has sides AB=4AB = 4, BC=6BC = 6, CA=8CA = 8. △DEF\triangle DEF has sides DE=6DE = 6, EF=9EF = 9, FD=12FD = 12. Are the triangles similar?
1

Sort each triangle's sides from shortest to longest

△ABC\triangle ABC: 4,6,84, 6, 8. △DEF\triangle DEF: 6,9,126, 9, 12 → Pairs: 4↔64 \leftrightarrow 6, 6↔96 \leftrightarrow 9, 8↔128 \leftrightarrow 12
2

Divide each side of △DEF\triangle DEF by the matching side of △ABC\triangle ABC

64=1.5\frac{6}{4} = 1.5, 96=1.5\frac{9}{6} = 1.5, 128=1.5\frac{12}{8} = 1.5 → All three ratios are 1.51.5
3

Apply SSS Similarity

Three equal ratios, so the sides are proportional → △ABC∼△DEF\triangle ABC \sim \triangle DEF, k=1.5k = 1.5

Common Mistakes

Pairing the sides in the order they are listed, for example 4,6,84, 6, 8 with 12,9,612, 9, 6, and getting ratios 33, 1.51.5, 0.750.75

Why it's wrong: The order in which side lengths are written says nothing about which sides correspond. Wrong pairs give different ratios even for similar triangles.

Correct: Sort both triangles from shortest to longest first: 4,6,84, 6, 8 and 6,9,126, 9, 12. Then all ratios are 1.51.5.

Using SAS with an angle that is not between the two sides

Why it's wrong: Two proportional sides and a non-included angle (SSA) do not fix the shape: two different triangles can have the same two sides and the same angle opposite one of them.

Correct: Check that the congruent angle is formed by exactly the two sides you compared. Otherwise SAS Similarity does not apply.

Thinking the area ratio is also kk

Why it's wrong: Area is length times length, so both factors grow by kk: the area grows by k⋅k=k2k \cdot k = k^2.

Correct: Perimeter ratio =k= k, area ratio =k2= k^2. With k=3k = 3, the area is 99 times as large, not 33 times.

Dividing big by small for one pair and small by big for another, for example 64=1.5\frac{6}{4} = 1.5 but 69≈0.67\frac{6}{9} \approx 0.67

Why it's wrong: The ratios then look different although the sides are proportional.

Correct: Always divide in the same direction: image side over original side (or the other way round for every pair).

Interactive Visual

Similar triangles

468ABC6912DEF△ABC ∼ △DEF

SSSAll three pairs of sides are in the same ratio k.

  • DE : AB = 6 : 4 = 1.5
  • EF : BC = 9 : 6 = 1.5
  • FD : CA = 12 : 8 = 1.5

Move the slider to change the scale factor k. The angles stay the same, every side is multiplied by k.

Similar triangles

58PQR1016XYZ△PQR ∼ △XYZ

SASTwo pairs of sides are in the same ratio k and the angles between them are equal.

  • XY : PQ = 10 : 5 = 2
  • YZ : QR = 16 : 8 = 2
  • ∠Q = ∠Y = 60°

Move the slider to change the scale factor k. The angles stay the same, every side is multiplied by k.

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which triangle is similar to a triangle with sides 22, 33 and 44?

Why It Matters

Often you can measure lengths more easily than angles, and then SSS and SAS similarity are the tools you need:
  • Surveying: to find the width of a lake you cannot cross, surveyors stake out a small triangle that shares an angle with a big one and use SAS similarity
  • Engineering: a model of a roof truss or a bridge is built with all lengths scaled by the same factor, so it has exactly the same shape as the real one
  • Printing and design: when a logo is enlarged, every length is multiplied by kk but the ink or material needed grows by k2k^2
  • Checking a drawing: a carpenter can check that a triangular frame matches a plan by comparing three length ratios, no protractor needed
Knowing that area grows with k2k^2 (not kk) also prevents costly mistakes, like ordering twice the paint for a wall that is twice as wide and twice as tall.

Real World Applications

Enlarging a Logo

Designers enlarge logos so that every length grows by the same scale factor. The enlarged triangle is similar to the original by SSS Similarity, so it keeps exactly the same shape.

Example:

A triangular logo with sides 22 cm, 33 cm and 44 cm is printed with k=5k = 5 on a poster: the sides become 1010 cm, 1515 cm and 2020 cm.

1Try It Yourself

A triangular logo has sides 33 cm, 44 cm and 66 cm. On a banner its shortest side must be 7.57.5 cm, and the logo must keep its shape.

How long is the longest side of the logo on the banner (in cm)?

Step 1: Write the mathematical expression

Multiply the longest side by the scale factor:

Measuring Across a Lake

Surveyors cannot stretch a tape measure across water. They mark two points on lines leading away from a corner point and use the small triangle that shares the corner angle with the big one.

Example:

From point AA two straight paths lead past the lake. A small triangle with sides 14\frac{1}{4} as long as the big one along both paths shares the angle at AA, so the width of the lake is 44 times the short distance measured on land.

2Try It Yourself

From point AA, AD=20AD = 20 m and AB=60AB = 60 m along one path, AE=25AE = 25 m and AC=75AC = 75 m along the other. BB and CC lie on opposite shores of the lake. On land, DE=18DE = 18 m.

How wide is the lake, the distance BCBC (in m)?

Step 1: Write the mathematical expression

Multiply the measured distance DEDE by the scale factor:

Fabric for a Bigger Sail

Sailmakers scale a proven sail design up or down. The new sail is similar to the old one, but the fabric needed grows with the area ratio $k^2$, not with $k$.

Example:

A sail that is 1.51.5 times as long on every side needs 1.52=2.251.5^2 = 2.25 times as much fabric.

3Try It Yourself

A small triangular sail has sides 33 m, 44 m and 55 m and an area of 66 m². A new sail has sides 66 m, 88 m and 1010 m.

How much fabric (in m²) does the new sail need?

Step 1: Write the mathematical expression

Multiply the old area by the area ratio:

Key Takeaways

  • 1SSS Similarity: if all three pairs of corresponding sides have the same ratio kk, the triangles are similar
  • 2SAS Similarity: two pairs of sides with the same ratio kk and a congruent included angle make the triangles similar
  • 3When only lengths are given, sort the sides from shortest to longest and pair them in that order
  • 4The scale factor kk is image side divided by original side; every missing side is kk times its partner
  • 5Perimeter ratio =k= k, area ratio =k2= k^2
  • 6Choose the criterion from what you know: two angles means AA, three sides means SSS, two sides and the angle between them means SAS

Frequently Asked Questions

SSS Congruence needs three pairs of equal sides. SSS Similarity only needs three pairs of sides in the same ratio. Congruence is the special case k=1k = 1.
SSS Congruence needs three pairs of equal sides. SSS Similarity only needs three pairs of sides in the same ratio. Congruence is the special case k=1k = 1.
The two sides and the angle between them fix the shape completely. If the angle is somewhere else (SSA), two different triangles can fit the same data, so you cannot conclude that the triangles are similar.
Look at what is given. Two pairs of congruent angles: AA. Three pairs of side lengths: SSS. Two pairs of side lengths and the angle between them: SAS. In nested triangles a shared angle often gives you the angle for AA or SAS for free.
Only for the value of kk. Big over small gives k>1k > 1 (enlargement), small over big gives k<1k < 1 (reduction). Both work, as long as you divide in the same direction for every pair.

Glossary

SSS Similarity
If all three pairs of corresponding sides of two triangles are in the same ratio, the triangles are similar
SAS Similarity
If two pairs of corresponding sides are in the same ratio and the included angles are congruent, the triangles are similar
Included angle
The angle formed by two given sides of a triangle, at the vertex where they meet
Scale factor
The ratio kk of a side of one similar figure to the corresponding side of the other
Corresponding sides
Sides that are in the same position in two similar or congruent figures
Area ratio
The ratio of the areas of two similar figures, equal to k2k^2

Formula Card

SSS Similarity

DEAB=EFBC=FDCA=k⇒△ABC∼△DEF\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k \Rightarrow \triangle ABC \sim \triangle DEF

All three side ratios are equal (sides sorted from shortest to longest)

SAS Similarity

DEAB=EFBC=k, ∠B=∠E⇒△ABC∼△DEF\frac{DE}{AB} = \frac{EF}{BC} = k,\ \angle B = \angle E \Rightarrow \triangle ABC \sim \triangle DEF

Two side ratios are equal and the included angles are congruent

Perimeter ratio

P2P1=k\frac{P_2}{P_1} = k

Perimeters grow by the same factor as the sides

Area ratio

A2A1=k2\frac{A_2}{A_1} = k^2

Areas grow by the square of the scale factor

Today's challengeOne problem, three tries

More in This Topic

Introduction to Congruence

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Introduction to Similarity

Intermediate25 min

Congruent Triangles

Intermediate25 min

AA Similarity

Intermediate25 min