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