Almost in every single European league you will see the players wear a badge to the right arm sleeve with the name and the logo of the League in which they play.
Same holds for international matches in the Europa League as well as in the Champions League. The badges are a way to promote the league sponsor as well as to promote the league itself which is also a commercial product vying for eyeballs, sponsors, and TV audience + advertisers.
The design of the badges vary highly though. In most leagues there is just one patch to the right arm sleeve. However, in England there is one on each sleeve. In Germany there is a league badge to the right arm and a sponsor logo to the left arm.
Usually there is a regular badge + a champs badge for the reigning champion.
A special badge is also available for the winner of the FIFA World Club championship.
Among recent winners are FC Barcelona, Corinthians, and FC Bayern Munich.
The shape of the badge is like a shield. 3D Senscilia flock material.
Dimensions: 5.6 cm across and 7.6 cm in length.

World Club Champions 2011 badge

 
In the UEFA Champions League the StarBall badge is round.
Dimensions: 7.5 cm diameter.

UEFA StarBall badge

 
The winners has the right and the obligation to wear a special Champs badge as can be seen below. Same shape as the regular StarBall badge but with a different design.
Dimensions: 7.5 cm diameter.

UCL winners badge 2013 FCB

The UEFA also has a different kind of badge. The Honors badge. A Club which lives up two either criteria may wear this on the left arm sleeve: Has won 3 consecutive Champions League titles. Has won a total of 4 Champions League title.
The shape of the badge is oval.
Dimensions: 5.8 cm across and 7.5 cm in length.

UEFA honors badge 5 Cups FCB

LFP sleeve badge for the Spanish League.
The shape of the badge is rectangular. The material is embroidery.
The replica badge as seen below as the dimensions: 5.2 cm across & 7.2 cm top down.
The authentic player’s badge as seen on TV measures: 7 cm X 10 cm.

LFP league badge for the Spanish La Liga

In the English Premier League the winner gets a special gold patch.
The shape of the badge is hard to describe with a single word but maybe something along the lines of a pentagon. The material is 3D Senscilia.
There are two sizes available. Player’s size and replica size. Below the player’s size badge is on display.
Dimensions at widest margin: 9.5 cm across + 8.5 cm top down.

Premier League Champs badge 12/13

The Premier League also has a standard league badge.
The dimensions: 9.4 cm across and 8.6 cm crown to point.
There are two sizes available. Player’s size and replica size. Below the player’s size badge is on display.

Premier League sleeve badge PRO S

In Germany they have a regular badge + a Champs badge.
The champs badge is in 3D Lextra with gold.
Dimensions: 6.5 cm across and 10 cm height.

Bundesliga Meister badge gold

The regular badge is in regular lextra and in red and white.
Dimensions: 6.2 cm across and 7.5 cm in height.

Bundesliga sleeve badge 50 years

In Germany they also allow former champs to display their honors by running a star scheme. 1 star for 3 Bundesliga titles. 2 stars for 5 BL titles. 3 stars for 10 BL titles. And for FC Bayern which plays in a league of its’ own there is also the option of 4 stars for 20 Bundesliga titles.
Dimensions: 1.8 cm point to point.

FC Bayern 4 stars above the logo

In Italy there is multitude of different badges available.
The winner of the league title gets the Scudetto badge.
The badge is embroidered and it comes with a golden edge. The shape of the badge is like a shield. It is placed to the chest and not at the sleeve.
Dimensions: 5 cm across and 6.8 cm top down.

Scudetto badge the Italian Serie A

 
The Lega Calcio sleeve badge to the right arm sleeve is the same to all Italian clubs in the Serie A though. The sponsor is telecommunications giant TIM.
Dimensions: 5.8 cm across and 7.2 cm length.

Lega Calcio sleeve badge Serie A

Any club in Italy which has won a total of 10 league titles is entitled to wear a star above the crest. 20 league titles gives the right to 2 stars.
Dimensions: 2.4 cm point to point of the star with widest possible margin.
(the AC Milan club crest measures 7.8 cm in length).

10 Scudetto titles 1 star above crest

 
In France there is one badge for the winner of the Ligue 1 title. And one badge for the rest of the teams in the first flight.
The badge looks to be mostly rectangular but with a curved end at the top.
Dimensions: 7 cm across and 10 cm top down.

France Ligue 1 sleeve badge

The Ligue 1 Champs badge is reserved for the winning club and thus is not available for general re-sale. Therefore we cannot offer a close-up view of the badge.
Not sure about the dimensions here ?

Ligue 1 Champs badge

The badge seems to be round with 3 stars on top. Not sure whether the 3 stars refer specifically to PSG or they are also present on the badge should another club win Le Championnat.
Red , blue and white. The colors from le Tri-Colore are present as well as Gold. Could not be any different could it ?
and what are the options exactly when trying to find the right shape for a badge ?

Geometric design shapes

 

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