Formulas

Elementary Geometry

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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.
    5. Right circular cylinder
    6. Lateral surface area of a right circular cylinder
      \({S_B} = 2\pi RH\)
    7. Total surface area of a right circular cylinder
      \(S = {S_L} + 2{S_B} \) \(= 2\pi R\left( {H + R} \right)\)
    8. Volume of a right circular cylinder
      \(V = {S_B}H \) \(= \pi {R^2}H\)
    9. 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 \({h_2}\).
    10. Truncated right circular cylinder
    11. Lateral surface area of a truncated cylinder
      \({S_L} = \pi R\left( {{h_1} + {h_2}} \right)\)
    12. Area of the bases of a truncated cylinder
      \({S_B} = \pi {R^2} \) \(+\;\pi R\sqrt {{R^2} + {{\left( {{\large\frac{{{h_1} – {h_2}}}{2}}\normalsize} \right)}^2}} \)
    13. Total surface area of a truncated cylinder
      \(S = {S_L} + {S_B} =\) \( \pi R\Big[ {{h_1} + {h_2} + R }\) \(+\;{ \sqrt {{R^2} + {{\left( {{\large\frac{{{h_1} – {h_2}}}{2}}\normalsize} \right)}^2}} } \Big]\)
    14. Volume of a truncated cylinder
      \(V = {\large\frac{{\pi {R^2}\left( {{h_1} + {h_2}} \right)}}{2}\normalsize}\)