Why is the FIRST logo a triangle, a circle, and a square?
Is there any story behind it?
A little bit of searching (aided by the knowledge that Jack Kamen–Dean’s father–designed the original logo) turned up this in one of the results. Watch the video.
TLDR version:
Archimedes’ theorem that the volume of a cone plus the volume of a sphere equals the volume of a cylinder (all of diameter d and, if appropriate, height d) inspired the triangle, circle, and square.
The colors are the national colors of the USA. (Recall that FIRST was originally (and still is officially) the United States Foundation for Inspiration and Recognition of Science and Technology.