Editing Frustum (section)

===Volume===
The formula for the volume of a pyramidal square frustum was introduced by the ancient [[Egyptian mathematics]] in what is called the [[Moscow Mathematical Papyrus]], written in the [[13th dynasty]] ({{circa|1850 BC}}):
:<math>V = \frac{h}{3}\left(a^2 + ab + b^2\right),</math>
where {{mvar|a}} and {{mvar|b}} are the base and top side lengths, and {{mvar|h}} is the height.

The Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.

The [[volume]] of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":
:<math>V = \frac{h_1 B_1 - h_2 B_2}{3},</math>
where {{math|''B''<sub>1</sub>}} and {{math|''B''<sub>2</sub>}} are the base and top areas, and {{math|''h''<sub>1</sub>}} and {{math|''h''<sub>2</sub>}} are the perpendicular heights from the apex to the base and top planes.

Considering that
:<math>\frac{B_1}{h_1^2} = \frac{B_2}{h_2^2} = \frac{\sqrt{B_1B_2}}{h_1h_2} = \alpha,</math>
the formula for the volume can be expressed as the third of the product of this proportionality, <math>\alpha</math>, and of the [[Factorization#Sum/difference of two cubes|difference of the cubes]] of the heights {{math|''h''<sub>1</sub>}} and {{math|''h''<sub>2</sub>}} only:
:<math>V = \frac{h_1 \alpha h_1^2 - h_2 \alpha h_2^2}{3} = \alpha\frac{h_1^3 - h_2^3}{3}.</math>
By using the identity {{math|1=''a''<sup>3</sup> − ''b''<sup>3</sup> = (''a'' − ''b'')(''a''<sup>2</sup> + ''ab'' + ''b''<sup>2</sup>)}}, one gets:
:<math>V = (h_1 - h_2)\alpha\frac{h_1^2 + h_1h_2 + h_2^2}{3},</math>
where {{math|1=''h''<sub>1</sub> − ''h''<sub>2</sub> = ''h''}} is the height of the frustum.

Distributing <math>\alpha</math> and substituting from its definition, the [[Heronian mean]] of areas {{math|''B''<sub>1</sub>}} and {{math|''B''<sub>2</sub>}} is obtained:
:<math>\frac{B_1 + \sqrt{B_1B_2} + B_2}{3};</math>
the alternative formula is therefore:
:<math>V = \frac{h}{3}\left(B_1 + \sqrt{B_1B_2} + B_2\right).</math>
[[Heron of Alexandria]] is noted for deriving this formula, and with it, encountering the [[imaginary unit]]: the square root of negative one.<ref>Nahin, Paul. ''An Imaginary Tale: The story of {{sqrt|−1}}.'' Princeton University Press. 1998</ref>

In particular:
*The volume of a circular cone frustum is:
::<math>V = \frac{\pi h}{3}\left(r_1^2 + r_1r_2 + r_2^2\right),</math>
:where {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are the base and top [[Radius (geometry)|radii]].

*The volume of a pyramidal frustum whose bases are regular {{mvar|n}}-gons is:
::<math>V = \frac{nh}{12}\left(a_1^2 + a_1a_2 + a_2^2\right)\cot\frac{\pi}{n},</math>
:where {{math|''a''<sub>1</sub>}} and {{math|''a''<sub>2</sub>}} are the base and top side lengths.
:[[Image:Frustum with symbols.svg|right|Pyramidal frustum|frameless]]