Theorem: In any circles the same ratio of arc length to radius determines a unique central angle that the arcs subtend;
and conversely, equal central angles determine the same ratio of arc length to radius.
Prove the theorem above using the definition of the radian measure of an angle.

Zed is making a centerpiece for his dining room table by putting sand in a cube-shaped vase. the vase is 0.25 meters in length, width, and height. he bought 25 kg of sand. assume the density of dry sand is 1,602 kgm3. what is the volume of the sand that zed purchased? the volume of the sand is approximately m3. enter your answer as the number that correctly fills in the blank in the previous sentence. if necessary, round your answer to six decimal places, like so: 42.536475