Group your triangles. Select that group and the circle. Goto Object > Clipping Mask > Make. Be sure that nothig else is selected and the circle is above the group. You may notice that when your mouse hovers over the vanished parts, Illustrator occasionally show the vanished parts with the used layer registration color. But that's only in Illustrator.
If that's too weird, you can use the shape builder and recreate the wanted areas by filling them with colors. Or you can lock everything else and divide with the circle objects below. Select only the circle and apply Object > Path > Divide Objects Below. In both cases you may need to ungroup and manually delete the remnants.
Here's an example of the division. All triangles are closed paths.Separate line segments will not be divided.
If you have grouped all triangles and used the circle as a clipping mask, you see now and then ghosts like this:
They do not any harm, they only rememind you that here is a masked shape.
Clipping mask doesn't create a stroke along the clipping border. The other presented methods create one and that can be harmful.