Can I use 10 AWG on a 45 amp dryer breaker?

No. NEC 240.4(D)(7) limits overcurrent protection for 10 AWG copper to 30 amperes, and its 35 ampere ampacity would only permit a 40 ampere device under 240.4(B).

Does the maximum breaker on a nameplate have to be used?

No. It is a maximum. NEC 422.11(E) prohibits exceeding it, but a smaller standard device that carries the load is acceptable.

Does a dryer circuit need four wires?

Yes for new work. NEC 250.140 requires a separate equipment grounding conductor, so two hots, a neutral, and a ground.

What wire size for a dryer rated 32.5 amps with a 45 amp breaker?

8 AWG copper. The nameplate reads 32.5 amperes, 240 volts, 45 ampere maximum overcurrent device, and 8 AWG copper at the 75 degrees C column of Table 310.16 is rated 50 amperes. That covers the 45 ampere breaker and the 32.5 ampere load with room to spare.

Why 10 AWG does not work here

10 AWG copper is 35 amperes at 75 degrees C, which comfortably exceeds the 32.5 ampere load. The problem is the breaker. Section 240.4 requires conductors to be protected at their ampacity, and 240.4(B) permits the next higher standard overcurrent device rating from 240.6(A) where the ampacity does not correspond to a standard size. The next standard size above 35 is 40, not 45, so a 45 ampere breaker over 10 AWG copper is not permitted.

Section 240.4(D)(7) closes the door entirely. Unless specifically permitted in 240.4(E) or 240.4(G), the overcurrent protection for 10 AWG copper shall not exceed 30 amperes. Article 422 appliance circuits are not on the 240.4(G) list, so 30 amperes is a hard ceiling for 10 AWG on this load.

Reading the nameplate correctly

Section 422.10(A) requires the branch circuit rating for an individual branch circuit supplying an appliance to be not less than the marked rating of the appliance. Section 422.11(E) then governs the overcurrent device. Where the appliance is marked with a maximum overcurrent protective device rating, the branch circuit device shall not exceed that rating. Note the word maximum. A 45 ampere marking is a ceiling, not an instruction.

You may use a smaller standard device as long as it carries the load. A 40 ampere breaker on 8 AWG copper is a perfectly valid choice for a 32.5 ampere dryer and is often the easier find at the counter. Both 40 and 45 are standard sizes under 240.6(A).

A clothes dryer is not a continuous load in the NEC sense, since it does not operate for three hours or more, so you do not apply the 125 percent factor from 210.19(A)(1) to the branch circuit conductors.

Receptacle, cord, and terminations

Section 250.140 requires new dryer installations to use a 4 wire connection with a separate equipment grounding conductor. Use a NEMA 14-30 or 14-50 configuration matched to the breaker per 210.21(B)(1) and Table 210.21(B)(3). A 45 ampere breaker with a 14-50 receptacle is acceptable, since 210.21(B)(3) allows a receptacle rating not less than the circuit rating for a single receptacle on an individual branch circuit.

Terminations follow 110.14(C)(1)(a). For circuits rated 100 amperes or less and conductors 14 AWG through 1 AWG, use the 60 degrees C column unless the equipment terminations are listed for 75 degrees C. Modern breakers and dryer terminal blocks are almost always marked 75 degrees C, but check. 8 AWG copper at 60 degrees C is 40 amperes, which would push you to a 40 ampere breaker if the terminals are 60 degrees C rated.

Field takeaway

8 AWG copper THHN or THWN-2, a 40 or 45 ampere two pole breaker, four conductors including a separate ground, and a NEMA 14 series receptacle. If the run is long, check voltage drop, and if the equipment terminals are only 60 degrees C rated, stay at 40 amperes.