Similarity ratio - Examples, Exercises and Solutions

What is the similarity ratio?

The similarity ratio is the constant difference between the corresponding sides of the two shapes.
That is, if the similarity ratio is 3 3 , we know that each side of the large triangle is 3 3 times larger than that of the small triangle.

How do we calculate the similarity ratio?

The calculation of the similarity ratio is divided into several steps that must be performed:

  1. First we must know that we are dealing with similar triangles or polygons.
  2. We must know how to identify the corresponding sides in each of the triangles or polygons.
  3. We need to know the sizes of a pair of equal sides.
  4. We must divide the size of one side by the size of the other side.

The result obtained is actually the similarity ratio.

Suggested Topics to Practice in Advance

  1. Similar Triangles
  2. Triangle similarity criteria
  3. Similarity of Triangles and Polygons

Practice Similarity ratio

Exercise #1


Choose the correct answer.

Video Solution

Step-by-Step Solution

First, let's look at angles C and E, which are equal to 30 degrees.

Angle C is opposite side AB and angle E is opposite side BD.

ABDB \frac{AB}{DB}

Now let's look at angle B, which is equal to 90 degrees in both triangles.

In triangle ABC the opposite side is AC and in triangle EBD the opposite side is ED.

ACED \frac{AC}{ED}

Let's look at angles A and D, which are equal to 60 degrees.

Angle A is the opposite side of CB, angle D is the opposite side of EB

CBEB \frac{CB}{EB}

Therefore, from this it can be deduced that:

ABBD=ACED \frac{AB}{BD}=\frac{AC}{ED}

And also:

CBED=ABBD \frac{CB}{ED}=\frac{AB}{BD}


Answers a + b are correct.

Exercise #2

AAABBBCCCMMMNNN36 What is the ratio between the sides of the triangles ΔABC and ΔMNA?

Video Solution

Step-by-Step Solution

From the data in the drawing, it seems that angle M is equal to angle B

Also, angle A is an angle shared by both triangles ABC and AMN

That is, triangles ABC and AMN are similar respectively according to the angle-angle theorem.

According to the letters, the sides that are equal to each other are:

ABAM=BCMN=ACAN \frac{AB}{AM}=\frac{BC}{MN}=\frac{AC}{AN}

Now we can calculate the ratio between the sides of the given triangles:

MN=3,BC=6 MN=3,BC=6 63=2 \frac{6}{3}=2


BCMN=2 \frac{BC}{MN}=2

Exercise #3

Is the similarity ratio between the three triangles equal to one?

Step-by-Step Solution

To answer the question, we first need to understand what "similarity ratio" means.

In similar triangles, the ratio between the sides is constant.

In the statement, we do not have data on any of the sides.

However, a similarity ratio of 1 means that the sides are exactly the same size.

That is, the triangles are not only similar but also congruent.

In the drawing, you can clearly see that the triangles are of different sizes and, therefore, clearly the similarity ratio between them is not 1.



Exercise #4

In the image there are a pair of similar triangles and a triangle that is not similar to the others.

Determine which are similar and calculate their similarity ratio.


Step-by-Step Solution

Triangle a and triangle b are similar according to the S.S.S (side side side) theorem

And the relationship between the sides is identical:

GHDE=HIEF=GIDF \frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}

96=31=62=3 \frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3

That is, the ratio between them is 1:3.


a a and b b , similarity ratio of 3 3

Exercise #5


The triangles above are similar.

Calculate the perimeter of the larger triangle.

Video Solution

Step-by-Step Solution

We calculate the perimeter of the smaller triangle (top):

3.5+1.5+4=9 3.5+1.5+4=9

Due to their similarity, the ratio between the sides of the triangles is equal to the ratio between the perimeters of the triangles.

We will identify the perimeter of the large triangle using x x :

x9=143.5 \frac{x}{9}=\frac{14}{3.5}

3.5x=14×9 3.5x=14\times9

3.5x=126 3.5x=126

x=36 x=36



Exercise #1

The similarity ratio between two similar triangles is 7, so that the area ratio is —— _{——}

Video Solution

Step-by-Step Solution

We square it. 7 squared is equal to 49.



Exercise #2

Here are two similar triangles. The ratio of the lengths of the sides of the triangle is 3:4, what is the ratio of the areas of the triangles?


Video Solution

Step-by-Step Solution

Let's call the small triangle A and the large triangle B, let's write the ratio:

AB=34 \frac{A}{B}=\frac{3}{4}

Square it:

SASB=(34)2 \frac{S_A}{S_B}=(\frac{3}{4})^2

SASB=916 \frac{S_A}{S_B}=\frac{9}{16}

Therefore, the ratio is 9:16



Exercise #3

5.213125 The triangle above are similar.

What is the perimeter of the blue triangle?

Video Solution

Step-by-Step Solution

The perimeter of the left triangle: 13+12+5=25+5=30

Therefore, the perimeter of the right triangle divided by 30 is equal to 5.2 divided by 13:

x30=5.213 \frac{x}{30}=\frac{5.2}{13}

13x=156 13x=156

x=12 x=12



Exercise #4

If the ratio of the areas of similar triangles is 1:16, and the length of the side of the larger triangle is 42 cm, what is the length of the corresponding side in the smaller triangle?

Video Solution

Step-by-Step Solution

The ratio of similarity is 1:4

The length of the corresponding side in the small triangle is:

424=6 \frac{42}{4}=6



Exercise #5

The triangle ABC is similar to the triangle DEF.

The ratio between the lengths of their sides is 9:8.

What is the ratio between the areas of the triangles?

Video Solution

Step-by-Step Solution

We multiply the ratio by 2

9:8=18:16 9:8=18:16

Raised to the power of 2:

92:82=81:64 9^2:8^2=81:64



Exercise #1

In similar triangles, the area of the triangles is 361 cm² and 81 cm². If it is known that the perimeter of the first triangle is 38, what is the perimeter of the second triangle?

Video Solution

Step-by-Step Solution

We note the reason for the perimeter according to the data as follows:

P2P1=S2S1 \frac{P_2}{P_1}=\sqrt{\frac{S_2}{S_1}}

We replace the existing data

P238=81361 \frac{P_2}{38}=\sqrt{\frac{81}{361}}

P238=81361=919 \frac{P_2}{38}=\frac{\sqrt{81}}{\sqrt{361}}=\frac{9}{19}

We multiply by 38

P2=919×38=18 P_2=\frac{9}{19}\times38=18



Exercise #2

ABCD is a parallelogram
BFCE is a deltoid


What is the area of the parallelogram ABCD?

Video Solution

Step-by-Step Solution

First, we must remember the formula for the area of a parallelogram:Lado x Altura \text{Lado }x\text{ Altura} .

In this case, we will try to find the height CH and the side BC.

We start from the side

First, let's observe the small triangle EBG,

As it is a right triangle, we can use the Pythagorean theorem (

A2+B2=C2 A^2+B^2=C^2 )

BG2+42=52 BG^2+4^2=5^2

BG2+16=25 BG^2+16=25

BG2=9 BG^2=9

BG=3 BG=3

Now, let's start looking for GC.

First, remember that the deltoid has two pairs of equal adjacent sides, therefore:FC=EC=9 FC=EC=9

Now we can also do Pythagoras in the triangle GCE.

GC2+42=92 GC^2+4^2=9^2

GC2+16=81 GC^2+16=81

GC2=65 GC^2=65

GC=65 GC=\sqrt{65}

Now we can calculate the side BC:

BC=BG+GT=3+6511 BC=BG+GT=3+\sqrt{65}\approx11

Now, let's observe the triangle BGE and DHC

Angle BGE = 90°
Angle CHD = 90°
Angle CDH=EBG because these are opposite parallel angles.

Therefore, there is a ratio of similarity between the two triangles, so:

HDBG=HCGE \frac{HD}{BG}=\frac{HC}{GE}

HDBG=7.53=2.5 \frac{HD}{BG}=\frac{7.5}{3}=2.5

HCEG=HC4=2.5 \frac{HC}{EG}=\frac{HC}{4}=2.5

HC=10 HC=10

Now that there is a height and a side, all that remains is to calculate.

10×11110 10\times11\approx110


110 \approx110

Exercise #3

Given that triangles ABC and DEF are similar, what is their ratio of similarity?


Video Solution



Exercise #4

BC is parallel to DE.

Fill in the gap:

AD=AEAC \frac{AD}{}=\frac{AE}{AC}


Video Solution



Exercise #5

According to which theorem are the triangles similar?

What is their ratio of similarity?


Video Solution


S.S.S., 12 \frac{1}{2}

Topics learned in later sections

  1. Similarity of Geometric Figures