Which statement correctly describes the number of radians in a circle?

Which statement correctly describes the number of radians in a circle?

• A. There are π radians in a circle
• B. There are 3.14 radians in a circle
• C. There are 4π radians in a circle
• D. There are 2π radians in a circle

Answer: (D)

Radians measure angles by comparing arc length to the circle’s radius. One radian is the angle that subtends an arc whose length equals the radius. For a full circle, the arc length is the circumference, which is 2πr. Dividing by the radius gives 2π radians for a complete turn, so a circle contains 2π radians (about 6.283 radians). This also lines up with degrees, since 360 degrees equals 2π radians and 180 degrees equals π radians. So the statement that there are 2π radians in a circle is the correct description. The other options mix up parts of a circle or use an approximation: π radians is about a semicircle, not a full circle; 4π radians would be two full revolutions; and 3.14 is just an approximate value for π, not the full-circle measure.

Radians measure angles by comparing arc length to the circle’s radius. One radian is the angle that subtends an arc whose length equals the radius. For a full circle, the arc length is the circumference, which is 2πr. Dividing by the radius gives 2π radians for a complete turn, so a circle contains 2π radians (about 6.283 radians). This also lines up with degrees, since 360 degrees equals 2π radians and 180 degrees equals π radians.

So the statement that there are 2π radians in a circle is the correct description. The other options mix up parts of a circle or use an approximation: π radians is about a semicircle, not a full circle; 4π radians would be two full revolutions; and 3.14 is just an approximate value for π, not the full-circle measure.