Frustum

Formulas

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 (c. 1850 BC):

${\displaystyle V=\frac{h}{3}\left(a^2+ab+b^2\right),}$

where a and b are the base and top side lengths, and 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":

${\displaystyle V=\frac{h_1{B}_1-h_2{B}_2}{3},}$

where B₁ and B₂ are the base and top areas, and h₁ and h₂ are the perpendicular heights from the apex to the base and top planes.

Considering that

${\displaystyle \frac{B_1}{h_1^2}=\frac{B_2}{h_2^2}=\frac{\sqrt{B_1{B}_2}}{h_1{h}_2}=\alpha ,}$

the formula for the volume can be expressed as the third of the product of this proportionality, ${\displaystyle \alpha }$, and of the difference of the cubes of the heights h₁ and h₂ only:

${\displaystyle V=\frac{h_1\alpha h_1^2-h_2\alpha h_2^2}{3}=\alpha \frac{h_1^3-h_2^3}{3}.}$

By using the identity a³ − b³ = (a − b)(a² + ab + b²), one gets:

${\displaystyle V=(h_1-h_2)\alpha \frac{h_1^2+h_1{h}_2+h_2^2}{3},}$

where h₁ − h₂ = h is the height of the frustum.

Distributing ${\displaystyle \alpha }$ and substituting from its definition, the Heronian mean of areas B₁ and B₂ is obtained:

${\displaystyle \frac{B_1+\sqrt{B_1{B}_2}+B_2}{3};}$

the alternative formula is therefore:

${\displaystyle V=\frac{h}{3}\left(B_1+\sqrt{B_1{B}_2}+B_2\right).}$

Heron of Alexandria is noted for deriving this formula, and with it, encountering the imaginary unit: the square root of negative one.

In particular:

• The volume of a circular cone frustum is:

${\displaystyle V=\frac{\pi h}{3}\left(r_1^2+r_1{r}_2+r_2^2\right),}$

where r₁ and r₂ are the base and top radii.

• The volume of a pyramidal frustum whose bases are regular n-gons is:

${\displaystyle V=\frac{nh}{12}\left(a_1^2+a_1{a}_2+a_2^2\right)\cot \frac{\pi }{n},}$

where a₁ and a₂ are the base and top side lengths.

Surface area

For a right circular conical frustum the slant height ${\displaystyle s}$ is

${\displaystyle {\displaystyle s=\sqrt{{\left(r_1-r_2\right)}^2+h^2},}}$

the lateral surface area is

${\displaystyle {\displaystyle \pi \left(r_1+r_2\right)s,}}$

and the total surface area is

${\displaystyle {\displaystyle \pi \left(\left(r_1+r_2\right)s+r_1^2+r_2^2\right),}}$

where r₁ and r₂ are the base and top radii respectively.

Examples

• On the back (the reverse) of a United States one-dollar bill, a pyramidal frustum appears on the reverse of the Great Seal of the United States, surmounted by the Eye of Providence.
• Ziggurats, step pyramids, and certain ancient Native American mounds also form the frustum of one or more pyramids, with additional features such as stairs added.
• Chinese pyramids.
• The John Hancock Center in Chicago, Illinois, is a frustum whose bases are rectangles.
• The Washington Monument is a narrow square-based pyramidal frustum topped by a small pyramid.
• The viewing frustum in 3D computer graphics is a virtual photographic or video camera's usable field of view modeled as a pyramidal frustum.
• In the English translation of Stanislaw Lem's short-story collection The Cyberiad, the poem Love and tensor algebra claims that "every frustum longs to be a cone".
• Buckets and typical lampshades are everyday examples of conical frustums.
• Drinking glasses and some space capsules are also some examples.
• Sound Catcher: a wooden sculpture in Lithuania.
• Valençay cheese
• Rolo candies
• Crème caramel, or flan