Volume of Frustum of Right Circular Cone

Theorem

Let $F$ be a frustum of a right circular cone.

The volume $\VV$ of $F$ is given as:

$\VV = \dfrac {\pi h \paren {a^2 + a b + b^2} } 3$

where:

$a$ and $b$ are the radii of the bases of $F$

$h$ is the altitude of $F$.

$\VV = \dfrac {h \paren {A_1 + A_2 + \sqrt {A_1 A_2} } } 3$

where:

$A_1$ and $A_2$ are the areas of the bases of $F$

$h$ is the altitude of $F$.

Here we have that $F$ be a frustum of a right circular cone.

Hence the bases of $F$ are circles.

From Area of Circle, the areas of the bases of $F$ are therefore:

$A_1 = \pi a^2$

$A_2 = \pi b^2$

Hence:

$\ds \VV$

$=$

$\ds \dfrac {h \paren {A_1 + A_2 + \sqrt {A_1 A_2} } } 3$

$\ds $

$=$

$\ds \dfrac {h \paren {\pi a^2 + \pi b^2 + \sqrt {\pi a^2 \pi b^2} } } 3$

$\ds $

$=$

$\ds \dfrac {\pi h \paren {a^2 + b^2 + a b} } 3$

after simplification

Hence the result.

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Frustum of Right Circular Cone of Radii $a, b$ and Height $h$: $4.42$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Frustum of Right Circular Cone of Radii $a, b$ and Height $h$: $7.42.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes