Question

there are exactly π radians in a semicircle. there are exactly dropdown π radians in a full circle. there are approximately dropdown radians in a full circle. 2π radians is equal to dropdown degrees. (image of a circle with center o, points p, k, j, q, t on the circumference, radii labeled r and 0.14r)

Explanation:

First Dropdown (There are exactly __π radians in a full circle):

A semicircle has \( \pi \) radians. A full circle is twice a semicircle, so the number of \( \pi \) radians in a full circle is \( 2 \) (since \( 2\times\pi = 2\pi \), which is the measure of a full circle in radians). So the first blank is filled with \( 2 \).

Second Dropdown (There are approximately __ radians in a full circle):

We know that \( \pi \approx 3.14 \), and a full circle is \( 2\pi \). So \( 2\times3.14 = 6.28 \). Thus, the approximate number of radians in a full circle is \( 6.28 \).

Third Dropdown ( \( 2\pi \) radians is equivalent to __ degrees):

We know that \( \pi \) radians is equal to \( 180^\circ \) (since a semicircle is \( 180^\circ \) and has \( \pi \) radians). So \( 2\pi \) radians is \( 2\times180^\circ = 360^\circ \), which is the measure of a full circle in degrees.

Final Answers:

• There are exactly \( \boldsymbol{2}\pi \) radians in a full circle.
• There are approximately \( \boldsymbol{6.28} \) radians in a full circle.
• \( 2\pi \) radians is equivalent to \( \boldsymbol{360} \) degrees.

Answer:

We know that \( \pi \) radians is equal to \( 180^\circ \) (since a semicircle is \( 180^\circ \) and has \( \pi \) radians). So \( 2\pi \) radians is \( 2\times180^\circ = 360^\circ \), which is the measure of a full circle in degrees.

Final Answers:

• There are exactly \( \boldsymbol{2}\pi \) radians in a full circle.
• There are approximately \( \boldsymbol{6.28} \) radians in a full circle.
• \( 2\pi \) radians is equivalent to \( \boldsymbol{360} \) degrees.