Plat vs. plan
The French words plat and plan both translate to the concept of being flat, level, or even, but they belong to different registers. The word plat is the standard, everyday term for something lacking bumps, depth, or elevation. In contrast, plan is a more technical, mathematical, or scientific adjective used to describe a perfectly level or planar surface.
Plat
The adjective plat is the everyday word used to describe something that is physically flat, level, or smooth, such as a landscape, an object without significant thickness, or a surface without slopes.
Le paysan cultive un terrain plat près de la rivière.
(The farmer cultivates a flat piece of land near the river.)
Elle a acheté un écran plat pour son nouveau salon.
(She bought a flat screen for her new living room.)
Nous avons marché sur un sentier plat pendant des heures.
(We walked on a flat trail for hours.)
Il préfère dormir sur un oreiller très plat pour soulager son cou.
(He prefers sleeping on a very flat pillow to relieve his neck.)
La mer est calme et la surface de l'eau est plate aujourd'hui.
(The sea is calm and the surface of the water is flat today.)
Ce pays est connu pour son paysage incroyablement plat.
(This country is known for its incredibly flat landscape.)
Plan
The adjective plan translates to flat, even, or planar, but it is primarily used in technical, scientific, geometric, or formal contexts to describe a perfectly two-dimensional or perfectly level surface.
Le géomètre vérifie que la surface est parfaitement plane.
(The surveyor checks that the surface is perfectly flat.)
Une figure plane possède uniquement deux dimensions.
(A planar figure has only two dimensions.)
L'ingénieur a dessiné une courbe sur une surface plane.
(The engineer drew a curve on a flat surface.)
L'artisan doit poncer le bois pour obtenir un côté plan.
(The craftsman must sand the wood to obtain a flat side.)
Ce miroir plan réfléchit la lumière sans déformer l'image.
(This flat mirror reflects light without distorting the image.)
L'expérience de physique nécessite un support plan et rigide.
(The physics experiment requires a flat and rigid support.)
Summary
To summarize, you should use plat for everyday objects, landscapes, and general situations where something lacks bumps, elevation, or thickness. You should reserve plan for technical, scientific, or mathematical contexts where a surface is formally described as geometrically flat, level, or planar.