Alphys' Self-Made Computer! [Part I]

Alphys, the Royal Scientist and is officially Undyne's "friend" , has been recently been designing a new computer by herself for research purposes. She is starting to make icons for her desktop. This is a symbol that she has used for the Core Expansion folder:

Upon inquiring more as to how she made the logo, she gave some more information...

• $a + b + c = 10$

• The $270°$ sector has the arc length of $7.5π$

• All angles that look perpendicular are perpendicular, and lines that look parallel are parallel

She then asked me to solve this question, as I was going to The Advanced Math Quiz | 20XX , and it was to happen in a week...

$\large \text{What is the perimeter of his figure? Dotted lines are not counted}$

$\large \text{Give your answer to two decimal places}$

$\color{#333333}\text{Whoever wants to submit a report for this problem can go here}$