# Ungula

In solid geometry, an **ungula** is a section or part of a solid of revolution, cut off by a plane oblique to its base.[1] A common instance is the spherical wedge. The term *ungula* refers to the hoof of a horse, an anatomical feature that defines a class of mammals called ungulates.

The volume of an ungula of a cylinder was calculated by Grégoire de Saint Vincent.[2] Two cylinders with equal radii and perpendicular axes intersect in four double ungulae.[3] The bicylinder formed by the intersection had been measured by Archimedes in The Method of Mechanical Theorems, but the manuscript was lost until 1906.

A historian of calculus described the role of the ungula in integral calculus:

- Grégoire himself was primarily concerned to illustrate by reference to the
*ungula*that volumetric integration could be reduced, through the*ductus in planum*, to a consideration of geometric relations between the lies of plane figures. The*ungula*, however, proved a valuable source of inspiration for those who followed him, and who saw in it a means of representing and transforming integrals in many ingenious ways.[4]^{:146}

## Cylindrical ungula

A cylindrical ungula of base radius *r* and height *h* has volume

- ,[5].

Its total surface area is

- ,

the surface area of its curved sidewall is

- ,

and the surface area of its top (slanted roof) is

- .

*Proof*

*Proof*

Consider a cylinder
bounded below by plane
and above by plane
where *k* is the slope of the slanted roof:

- .

Cutting up the volume into slices parallel to the *y*-axis, then a differential slice, shaped like a triangular prism, has volume

where

is the area of a right triangle whose vertices are, , , and , and whose base and height are thereby and , respectively. Then the volume of the whole cylindrical ungula is

which equals

after substituting .

A differential surface area of the curved side wall is

- ,

which area belongs to a nearly flat rectangle bounded by vertices , , , and , and whose width and height are thereby and (close enough to) , respectively. Then the surface area of the wall is

where the integral yields , so that the area of the wall is

- ,

and substituting yields

- .

The base of the cylindrical ungula has the surface area of half a circle of radius *r*:
, and the slanted top of the said ungula is a half-ellipse with semi-minor axis of length *r* and semi-major axis of length
, so that its area is

and substituting yields

- . ∎

Note how the surface area of the side wall is related to the volume: such surface area being , multiplying it by gives the volume of a differential half-shell, whose integral is , the volume.

When the slope *k* equals 1 then such ungula is precisely one eighth of a bicylinder, whose volume is
. One eighth of this is
.

## Conical ungula

A conical ungula of height *h*, base radius *r*, and upper flat surface slope *k* (if the semicircular base is at the bottom, on the plane *z* = 0) has volume

where

is the height of the cone from which the ungula has been cut out, and

- .

The surface area of the curved sidewall is

- .

As a consistency check, consider what happens when the height of the cone goes to infinity, so that the cone becomes a cylinder in the limit:

so that

- ,

- , and

- ,

which results agree with the cylindrical case.

*Proof*

*Proof*

Let a cone be described by

where *r* and *H* are constants and *z* and *ρ* are variables, with

and

- .

Let the cone be cut by a plane

- .

Substituting this *z* into the cone's equation, and solving for *ρ* yields

which for a given value of *θ* is the radial coordinate of the point common to both the plane and the cone that is farthest from the cone's axis along an angle *θ* from the *x*-axis. The cylindrical height coordinate of this point is

- .

So along the direction of angle *θ*, a cross-section of the conical ungula looks like the triangle

- .

Rotating this triangle by an angle
about the *z*-axis yields another triangle with
,
,
substituted for
,
, and
respectively, where
and
are functions of
instead of
. Since
is infinitesimal then
and
also vary infinitesimally from
and
, so for purposes of considering the volume of the differential trapezoidal pyramid, they may be considered equal.

The differential trapezoidal pyramid has a trapezoidal base with a length at the base (of the cone) of , a length at the top of , and altitude , so the trapezoid has area

- .

An altitude from the trapezoidal base to the point has length differentially close to

- .

(This is an altitude of one of the side triangles of the trapezoidal pyramid.) The volume of the pyramid is one-third its base area times its altitudinal length, so the volume of the conical ungula is the integral of that:

where

Substituting the right hand side into the integral and doing some algebraic manipulation yields the formula for volume to be proven.

For the sidewall:

and the integral on the rightmost-hand-side simplifies to . ∎

As a consistency check, consider what happens when *k* goes to infinity; then the conical ungula should become a semi-cone.

which is half of the volume of a cone.

which is half of the surface area of the curved wall of a cone.

### Surface area of top part

When , the "top part" (i.e., the flat face that is not semicircular like the base) has a parabolic shape and its surface area is

- .

When then the top part has an elliptic shape (i.e., it is less than one-half of an ellipse) and its surface area is

where

- ,

- ,

- ,

- , and

- .

When
then the top part is a section of a hyperbola and its surface area is

where

- ,

- is as above,

- ,

- ,

- ,

- ,

where the logarithm is natural, and

- .

## See also

## References

- Ungula at Webster Dictionary.org
- Gregory of St. Vincent (1647)
*Opus Geometricum quadraturae circuli et sectionum coni* - Blaise Pascal Lettre de Dettonville a Carcavi describes the onglet and double onglet, link from HathiTrust
- Margaret E. Baron (1969)
*The Origins of the Infinitesimal Calculus*, Pergamon Press, republished 2014 by Elsevier, Google Books preview - Solids - Volumes and Surfaces at The Engineering Toolbox

- William Vogdes (1861) An Elementary Treatise on Measuration and Practical Geometry via Google Books