I need help finding the orbital period of the moon.

The question doesn't make sense to me. I don't see how the angular area of the Moon gives any useful information about its period or how fast it revolves around Earth. If there was a different moon closer in with the same angular area it would be smaller and move faster.

If by "period", you mean "the time it takes the Moon to go around the Earth once", then it self-evidently has a period of 360 degrees.

I think the only way this makes sense as a problem is if this is part of an astronomy lab, where you are supposed to use a telescope (or maybe a sextant?) to measure the position of the Moon at two different times (at least two different times), using some close background star as a stationary reference and the known angular diameter of the Moon (close to half a degree across, I think). To be valid data the calculated period must be within a certain range of the known orbital period of the Moon.

The motion of the moon relative to the earth is a 5 degrees tilted ellipse . if the motion of the moon as view from a fixed star, its motion relalive to the Sun appear to be cyclod whose period is 27.3 days.
Thus the moon orbits the sun in about 13.4 cycloidal periods which are called side real months.
The Angular size of the moon is the same as that of the Sun