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Cylindrical and planar symmetries yield two of the most important results in electrostatics.

Infinite line charge (linear charge density lambda): Gaussian surface: coaxial cylinder of radius r and length L. The curved surface has E perpendicular to it (and constant), while E is parallel to the flat end caps (zero flux contribution). Gauss's law: E * 2pirL = lambdaL/epsilon_0. Result: E = lambda/(2piepsilon_0*r). The 1/r dependence is characteristic of cylindrical symmetry.

Uniformly charged infinite cylinder (volume charge density rho, radius R): Inside (r < R): E = rhor/(2epsilon_0). Field increases linearly. Outside (r > R): E = rhoR^2/(2epsilon_0r) = lambda/(2piepsilon_0r) where lambda = rhopiR^2. Same as a line charge.

Infinite plane sheet (surface charge density sigma): Gaussian surface: pillbox (thin cylinder) straddling the sheet, with cross-sectional area A. Only the two flat faces contribute flux: 2EA = sigmaA/epsilon_0. Result: E = sigma/(2epsilon_0), uniform and independent of distance. This remarkable result means the field is the same whether you are 1 mm or 1 km from an infinite sheet.

Two parallel sheets with opposite charges (+sigma, -sigma): Between the sheets, fields add: E = sigma/epsilon_0 (from + to -). Outside, fields cancel: E = 0. This creates the uniform field of a parallel plate capacitor.

Conductor surface: Using a pillbox with one face inside (E = 0) and one outside: EA = sigmaA/epsilon_0. Result: E = sigma/epsilon_0 (not sigma/2*epsilon_0, because the field exists only on one side).