According to Cornell University most farms in 1850 were under 200 acres. According to Auburn southern plantations were anywhere from 500 to 1,000 acres.

Most farmers weren't in it to get rich, their intent was to provide for their family and, hopefully, have a comfortable life. The, relatively, small size of the farm meant that it could be worked by the family or a couple of hired hands. Farming communities were built around small towns with a general store, blacksmith, butcher, etc, and markets where surplus crops or homemade goods could be bartered or sold. During the brief grain harvest season the community would pool resources by helping each other with the harvest. Additionally, mechanical horse drawn wheat harvesters had been around for a long time (first "modern" harvester in 1826, first combine in 1835, and first corn harvester in 1831). In short, you could run a family farm with just a few people and the farmer's family was responsible for domestic tasks.

Plantations owners were in it for profit. Plantations were run in a self contained manner. The plantation would have it's own gardeners, blacksmith, butcher, cleaning staff, cooks etc. The plantation owner was a "manager", he and his family lived a life that would have been familiar to any feudal lord.

Additionally, cotton & tobacco were labor intensive. Although the cotton gin (for processing cotton) was invented in 1793, the first mechanical harvester wasn't developed until 1850; even then cotton still had to be planted by hand and the cotton bolls still had to be sorted and stripped by hand.

Tobacco was even more labor intensive. Planting, pruning, transplanting, pest removal, etc all had to be done by hand. Each plant had to be harvested as it became mature and the picked leaves have to be tended to as they ripen. Mechanical planters and harvesters didn't come into use until after WWII.

Wheat and corn were, primarily, grown on family farms, centered around towns that provided needed services. Size, mechanization, and purpose meant that fewer people were needed to work the farm and manage the household.

Cotton and tobacco were very labor intensive and grown on huge, self contained estates with the goal of making money. The plantation owner was a manager who needed a large number of people to handle his crop and his household.

A follow up question, but this might be too general. Slavery seems to be deeply interwoven with the idea of the plantation/the plantation economy. I'm thinking, in particular, about the ways in which the emergence of Venetian, Genoese, and later Spanish sugar plantations coincided with the return of large scale agricultural slavery to Europe.

Has slavery or other forms of unfree labour been a necessary historical condition for the kind of 'cash crop' oriented plantation economy that OP describes?

It was in fact used for both staple agriculture and "cash crops." Most large plantations were in some way self sufficient enterprises, providing their own foodstuffs as well as cash crops. At least until the 1850's this trend persisted but a shift towards cotton monoculture did occur. From DOWN TO EARTH NATURE'S ROLE IN AMERICAN HISTORY by Ted Steinberg

"Apart from bad weather conditions, southerners themselves played a hand in bringing on the chronic shortage of food. No trend contributed more to the dilemma than the move toward cotton monoculture. Cotton specialization would take a giant step forward after the war. But before that, in the 18S0s, when the price of the white lint rose dramatically, planters invested heavily in the crop, so heavily that per capita production of such essential foods as corn and sweet potatoes fell over the decade. When the Civil War began, many southern agricul- tural leaders called upon people to show their support for the Confederacy by abandoning cotton and growing more corn. "Plant Corn and Be Free, or plant cotton and be whipped," one Georgia newspaper declared. Confederate soldiers, the paper went on to explain, "will be powerless against grim hunger and gaunt famine, such as will overwhelm us if we insanely raise Cotton instead of corn." It would not prove easy, however, for the planter elite to give up its money crop and risk the region's monopoly over world cotton production, especially with many believing that foreign dependence on southern cotton would help bring such powers as Britain into the war on their side."

However corn was the main crop grown alongside cotton, the phrase "cotton and corn" plantation is very common in historical literature of the era. But this combined with hogs as was common on Southern plantations are not nutritionally sufficient for human beings. This forced slaveowners to allow their slaves to cultivate small plots of variable produce for their own consumption, and to allow for trapping, hunting, and berry gathering (wild blackberries are very common in the southeast). This information is from The Republic of Nature by Mark Fiege.

Other sources though make note of the presence of multi crop agriculture. The black belt, the heart of plantation agriculture in the south, was known as the "Granary of the confederacy" during the war due to the amount of food and grains it produced.

https://aurora.auburn.edu/bitstream/handle/11200/2387/1636BULL.pdf;sequence=1

In fact, cotton production reached its largest height after the civil war during the early days of sharecropping as farms sought to take advantage of the high price of cotton. The amount of cotton grown during the time created massive erosion in many places resulting in a dramatic restructuring of the landscape, an example of this is the little Grand Canyon in Western Georgia.

"The locus of contention over common rights, and particularly over common grazing rights, shifted from the Black Belt to the Hill Country during the late1870sand 188Os, reflecting the advance of commercialag- riculture into areas previously dominated by semisubsistence farming. Indeed, the absorption of the Southern Hill, or Upcountry, into the cot- ton economy was one of the significant developments of the postwar period. Inhabitants of this region had been hard hit by wartime priva- tion, drought, and pillaging; destitutionwas widespread in 1865and for a while thereafter. Local officials of the Freedmen's Bureau spent much of their time distributing needed rations to whites, not blacks. By the first years of the next decade, however, a new orientation seemed evident. "Cotton," a county newspaper reported in 1872, "formerly cultivated on a very limited extent, has increased rapidly in the last few years in pro- duction...". And the trend would continue so that by century's end white labor raised most of the cotton in the South.61 The initial thrust toward market production likely came from yeomen fafmers hoping to avail themselves of relatively high cotton prices and recoup war-related losses. Railroads, which began to pene- trate the Upcountry during the 187Os,offered added inducement by eas- ing access to supra-local markets. But most importantly, the transition came as a product of new social relations and credit arrangements. Like many peasantries,Upcountry yeomen normally cultivateda limited cash crop along with foodstuffs; the dynamic of the household economy, itself, encouraged the pattern. Even in1873, a Black Belt paper could note that these farmers attended to “their grain, provender, and provisions and then bestowed their surplus labor on cotton.”62What turned cotton from one item in a crop mix into a defining feature of the region’s econ- omy was a process that tied yeomen firmly to the export market, made vulnerable their productive property, and eventually reduced many of them to tenants and laborers. Expanded lines of transportation and com- munication and legislation bearing on production and exchange laid the foundation for the transformation; supply merchants carried it through."

From Red Harvest TheCommunistParty andAmericanFarmers by Lowell K. Dyson