Formulas and Tables

Elementary Geometry

Cylinder

Radius of the base of a cylinder: \(R\)
Generatrix of a cylinder: \(L\)
Height of a cylinder: \(H\)
Heights of a truncated cylinder: \({h_1},\) \({h_2}\)

Area of the base: \({S_B}\)
Lateral surface area: \({S_L}\)
Total surface area: \(S\)
Volume: \(V\)

  1. A cylinder is a geometric solid bounded by a cylindrical surface and two parallel planes crossing it. The cylindrical surface is formed by a straight line (called the generatrix) moving parallel to itself, so that any fixed point of the line moves along a plane curve called the directrix.
  2. A cylinder is called a circular cylinder if its directrix is a circle.
  3. A cylinder is called a right cylinder if it generatrix is perpendicular to the bases.
  4. A right circular cylinder is determined by the radius of the base \(R\) and the generatrix \(L,\) which is equal to the height \(H\) of the cylinder.
A right circular cylinder
  1. Lateral surface area of a right circular cylinder
    \({S_B} = 2\pi RH\)
  2. Total surface area of a right circular cylinder
    \(S = {S_L} + 2{S_B} \) \(= 2\pi R\left( {H + R} \right)\)
  3. Volume of a right circular cylinder
    \(V = {S_B}H \) \(= \pi {R^2}H\)
  4. A truncated right circular cylinder or briefly a truncated cylinder is determined by the radius of the base \(R,\) the shortest height \({h_1}\) and the greatest height \({h_2}\).
A truncated right circular cylinder
  1. Lateral surface area of a truncated cylinder
    \({S_L} = \pi R\left( {{h_1} + {h_2}} \right)\)
  2. Area of the bases of a truncated cylinder
    \({S_B} = \pi {R^2} \) \(+\;\pi R\sqrt {{R^2} + {{\left( {{\large\frac{{{h_1} – {h_2}}}{2}}\normalsize} \right)}^2}} \)
  3. Total surface area of a truncated cylinder
    \(S = {S_L} + {S_B} =\) \( \pi R\Big[ {{h_1} + {h_2} + R }\) \(+\;{ \sqrt {{R^2} + {{\left( {{\large\frac{{{h_1} – {h_2}}}{2}}\normalsize} \right)}^2}} } \Big]\)
  4. Volume of a truncated cylinder
    \(V = {\large\frac{{\pi {R^2}\left( {{h_1} + {h_2}} \right)}}{2}\normalsize}\)