Question

What is the most common name for an angle measurer

Answer 1

The most common name is degrees but advanced mathmatitions  use radions

Answer 2

You said "measurer", so I think you're asking for the common name of
the device that's commonly used to measure the size of an angle.

The device is a half-circle or a full circle, usually made of transparent
plastic, with angles from zero to 180 or from zero to 360 marked around
its edge ... like a ruler bent into a circle, only with marks of degrees instead
of inches or centimeters.

The common name for that device is a "protractor".