Formulas and Tables

Elementary Geometry

Frustum of Cone

Radii of the bases: \(R\), \(r\)
Generatrix (slant height): \(m\)
Height: \(H\)
Volume: \(V\)

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

  1. A frustum of a cone is a portion of a cone lying between the base
    and a parallel plane cutting the cone.
  2. Most often, the term “frustum of a cone” refers to a frustum of a right circular cone. Such a truncated cone is formed by rotating a right-angled trapezium about its leg perpendicular to the bases of the trapezium. A frustum of a right circular cone is determined by the radii of the bases \(R\) and \(r\) and height \(H\) (or slant height \(m\)).
Frustum of Cone
  1. Relation between the height, radii of the bases and slant height in a frustum of a cone
    \(H = \sqrt {{m^2} – {{\left( {R – r} \right)}^2}} \)
  2. bases of a frustum of a cone are similar to each other:
    \({\large\frac{R}{r}\normalsize} = k\),
    where \(k\) is a scale factor.
  3. Ratio of the areas of the top and bottom bases
    \({\large\frac{{{S_2}}}{{{S_1}}}\normalsize} = {\large\frac{{{R^2}}}{{{r^2}}}\normalsize} = {k^2},\)
    where \(k\) is a scale factor.
  4. Lateral surface area of a frustum of a cone
    \({S_L} = \pi m\left( {R + r} \right)\)
  5. Total surface area of a frustum of a cone
    \(S = {S_1} + {S_2} + {S_L} =\) \(\pi \big[ {{R^2} + {r^2} + m\left( {R + r} \right)} \big]\)
  6. Volume of a frustum of a cone
    \(V =\) \({\large\frac{h}{3}\normalsize}\left( {{S_1} + \sqrt {{S_1}{S_2}} + {S_2}} \right)\)