# 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: \(S\)

- A frustum of a cone is a portion of a cone lying between the base

and a parallel plane cutting the cone. - 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 \(m\)).

- Relation between the height, radii of the bases and slant height in a frustum of a cone

\(H = \sqrt {{m^2} – {{\left( {R – r} \right)}^2}} \) - bases of a frustum of a cone are similar to each other:

\({\large\frac{R}{r}\normalsize} = k\),

where \(k\) is a scale factor. - 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. - Lateral surface area of a frustum of a cone

\({S_L} = \pi m\left( {R + r} \right)\) - Total surface area of a frustum of a cone

\(S = {S_1} + {S_2} + {S_L} =\) \(\pi \big[ {{R^2} + {r^2} + m\left( {R + r} \right)} \big]\) - Volume of a frustum of a cone

\(V =\) \({\large\frac{h}{3}\normalsize}\left( {{S_1} + \sqrt {{S_1}{S_2}} + {S_2}} \right)\)