Will these plates fit the dishwasher and the cupboard?

The shelf height, the depth of the cupboard, and the width of a dishwasher slot. Take them with a tape measure rather than guessing, because a difference of half an inch decides whether a stack goes in upright.

Write them down. They stay useful for every piece of tableware you buy afterwards.

Six plates stand 3.3–5.1 inches, six bowls 8.4–13.2 inches. Those are calculations from the stated sizes rather than measurements of ours, and they are deliberately given as a range: rim shapes differ.

If the shelf is tight, two stacks of three always fit where one stack of six does not.

Ten-inch plates fit most machines. The rectangular plates are the ones that catch people out, because it is the length rather than the diameter that has to clear the basket.

Measure the longest piece you are buying, not the average.

Buy the smaller size rather than fighting the cupboard. Eight-inch plates do most of what ten-inch plates do, take less shelf and cost less.

The other honest answer is to keep the big pieces out of the cupboard entirely: serving pieces live well on open shelves, where their size is a feature.

Cupboard depth. A shelf tall enough for a stack is useless if the door closes on a ten-inch plate that sits half an inch proud.

Measure from the back wall to the closed door, not to the front edge of the shelf.

Every other shape here is measured across. The rectangular plates are measured along, and that length is what has to clear a dishwasher basket and a cupboard shelf.

They also stack differently: flat pieces nest almost perfectly, so six of them take less height than six round plates of the same size. If the cupboard is the constraint rather than the table, they are the shape to buy.