Two jobs, one short word
In present-day English, plan names both a sequence of intended actions and a drawing of how something is laid out in space. You can plan a campaign, hold the plans for a house, or subscribe to a phone plan. The verb follows the noun: to plan is to set those arrangements down before the work begins. The senses feel close once you notice that each is an orderly arrangement toward an end. Historically they are close for a more concrete reason. English took the word from French in the seventeenth century, and French had already used it for maps, ground plots, and schemes.
The origin of the word plan is not a single Latin ancestor. Dictionaries that look tidy on the page are compressing a merger.
When English borrowed it
The loan is early modern, not medieval. It does not go back to Old English. Specialists disagree on the first secure date, mainly because early uses are scarce and sit in technical prose.
The Oxford English Dictionary, in its 2006 revision, puts a scheme-of-action sense in 1635, in a Scottish survey of sciences that discusses how modern armies are marshalled. A drawing sense appears soon after, in 1664, in a remark on Palladio’s collection of antique building layouts, plans set beside profiles. Merriam-Webster likewise dates the English noun to 1635. The Online Etymology Dictionary is more cautious. It treats the word as a technical term in perspective drawing from the 1670s, a general label for a sketch or diagram by 1706, and a formulated scheme of action only by 1713.
The verb is later and clearly derived. By the late 1720s it could mean putting the parts and dimensions of a building on paper. By the 1730s it could mean devising ways and means. Merriam-Webster’s first verb date is a little earlier, 1718. None of these figures needs to be forced into one year. What is stable is the path: a French architectural and cartographic term entered English, then widened, within a few decades, from paper layout to intended action.
Half flat surface, half planting
French plan itself is a blend, and the better references now say so.
One stream is plan as a plane surface, recorded in the mid-sixteenth century. That is Latin planum, a noun from the adjective planus, meaning level or flat. The deeper root is the Indo-European base for spreading and flattening, the same family that yields English plane and plain, Spanish plano and inherited llano, and Italian piano. On this account a plan is a drawing on a flat sheet: a ground plot, a map, a projection onto a horizontal plane. Etymonline tells this version almost alone. It is not false. It is incomplete.
The other stream is French plant, later spelled plan, for the drawing that guides how a building, town, or work is set on the ground. That form is attested as plant in the 1530s and 1540s and as plan by the 1540s and 1560s. It comes from planter, to plant or fix in place, from Latin plantare, built on planta, originally the sole of the foot. Italian pianta in the sense of a building or town design belongs here and likely reinforced the French architectural use. Spanish planta shows the same pattern. On this account a plan is first a planting, a fixing onto soil, and only then a drawing of that planting.
American Heritage treats English plan as French plant reshaped under the influence of plan meaning flat surface. Merriam-Webster and the OED say the English word is partly from each French source. A paper by Bernhard Diensberg has argued that ordinary English dictionaries underplay the plant stage. Historical notes on the French side add a practical detail: final t often dropped or assimilated, so plant and plan were easy to mix in the sixteenth century. Some French dictionaries still keep two lemmas. The historical dictionaries admit they were confused with each other.
The honest summary is the one language historians reach after sitting with the OED revision: English borrowed a word that was already half plane and half plant.
From a floor layout to a scheme
French had already stretched the architectural sense toward a set of measures adopted in order to get something done, by the 1620s. English followed the same slide. Once a word names the graphic layout of a building, it is a short step to the layout of a siege, a day’s work, or a policy. The drawing sense never died. Floor plan, city plan, and body plan still mean arrangement in space. In ordinary speech, though, the default became the scheme.
Later English uses are extensions of that default, not a new etymology. A method or customary procedure; a goal or aim; in the twentieth century, a detailed program of service or payment. A pension plan and an installment plan are packaged sequences, not sketches, but they still name an orderly arrangement of parts. Compounds such as game plan and master plan keep the strategic sense in view. None of this required a second borrowing.
Cognates track the split. German Plan is itself a French loan and covers project, map, and scheme in much the way English does. Catalan pla and a cluster of European learned forms sit with Latin planus. The Italian and Spanish pianta / planta line sits with planting and the sole of the foot. English plant is a relative of that second line, not a folk pun.
What the merger leaves you with
If you want one fact that the surface of the word does not advertise, keep the fork. Plan looks like plane and plain because, in part, it is: a later French form of Latin “flat.” It also looks like plant because, in part, it is that too: a ground-plot word from fixing something in place. English did not choose between those stories. It took the blend, then let the scheme sense outgrow the drawing.