Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

How many radians are in a full circle?

1. 1 radian

2. 2π radians

3. 360 radians

4. 180 radians

The correct answer is: 2π radians

A full circle has an angular measurement of \( 2\pi \) radians. This measurement is derived from the relationship between degrees and radians: there are \( 360 \) degrees in a full circle, and since one radian is defined as the angle subtended by an arc length equal to the radius of the circle, the conversion between degrees and radians is established as follows: 1 full circle in degrees \( = 360^\circ \) 1 full circle in radians \( = 2\pi \) radians. The mathematical formula to convert degrees to radians is to multiply the degree measure by \( \frac{\pi}{180} \). Thus, when you apply this conversion to \( 360^\circ \): \[ 360^\circ \times \frac{\pi}{180} = 2\pi \text{ radians}. \] This equivalence shows that a full circle indeed measures \( 2\pi \) radians, which is the correct answer. Understanding this relationship is essential in both geometry and trigonometry as it lays the foundation for measuring angles in various applications, including those encountered in physics and engineering.