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Further terminology

The perimeter of the base of a cone is called the directrix, and each of the line segments between the directrix and apex is a generatrix or generating line of the lateral surface. (For the connection between this sense of the term directrix and the directrix of a conic section, see Dandelin spheres.)

The base radius of a circular cone is the radius of its base; often this is simply called the radius of the cone. The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle θ to the axis, the aperture is 2θ. In optics, the angle θ is called the half-angle of the cone, to distinguish it from the aperture.

A cone with a region including its apex cut off by a plane is called a truncated cone; if the truncation plane is parallel to the cone's base, it is called a frustum.⁴ An elliptical cone is a cone with an elliptical base.⁵ A generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary (see Visual hull).

Measurements and equations

Volume

The volume V {\displaystyle V} of any conic solid is one third of the product of the area of the base A B {\displaystyle A_{B}} and the height h {\displaystyle h} ⁶

V = 1 3 A B h . {\displaystyle V={\frac {1}{3}}A_{B}h.}

In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral

∫ x 2 d x = 1 3 x 3 {\displaystyle \int x^{2}\,dx={\tfrac {1}{3}}x^{3}}

Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying Cavalieri's principle – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the method of exhaustion. This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are scissors congruent (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.⁷

Center of mass

The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.

Right circular cone

Volume

For a circular cone with radius r {\displaystyle r} and height h {\displaystyle h} , the base is a circle of area π r 2 {\displaystyle \pi r^{2}} thus the formula for volume is:⁸

V = 1 3 π r 2 h {\displaystyle V={\frac {1}{3}}\pi r^{2}h}

Slant height

The slant height of a right circular cone is the distance from any point on the circle of its base to the apex via a line segment along the surface of the cone. It is given by r 2 + h 2 {\displaystyle {\sqrt {r^{2}+h^{2}}}} , where r {\displaystyle r} is the radius of the base and h {\displaystyle h} is the height. This can be proved by the Pythagorean theorem.

Surface area

The lateral surface area of a right circular cone is L S A = π r ℓ {\displaystyle LSA=\pi r\ell } where r {\displaystyle r} is the radius of the circle at the bottom of the cone and ℓ {\displaystyle \ell } is the slant height of the cone.⁹ The surface area of the bottom circle of a cone is the same as for any circle, π r 2 {\displaystyle \pi r^{2}} . Thus, the total surface area of a right circular cone can be expressed as each of the following:

• Radius and height

π r 2 + π r r 2 + h 2 {\displaystyle \pi r^{2}+\pi r{\sqrt {r^{2}+h^{2}}}} (the area of the base plus the area of the lateral surface; the term r 2 + h 2 {\displaystyle {\sqrt {r^{2}+h^{2}}}} is the slant height) π r ( r + r 2 + h 2 ) {\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)} where r {\displaystyle r} is the radius and h {\displaystyle h} is the height.

• Radius and slant height

π r 2 + π r ℓ {\displaystyle \pi r^{2}+\pi r\ell } π r ( r + ℓ ) {\displaystyle \pi r(r+\ell )} where r {\displaystyle r} is the radius and ℓ {\displaystyle \ell } is the slant height.

• Circumference and slant height

c 2 4 π + c ℓ 2 {\displaystyle {\frac {c^{2}}{4\pi }}+{\frac {c\ell }{2}}} ( c 2 ) ( c 2 π + ℓ ) {\displaystyle \left({\frac {c}{2}}\right)\left({\frac {c}{2\pi }}+\ell \right)} where c {\displaystyle c} is the circumference and ℓ {\displaystyle \ell } is the slant height.

• Apex angle and height

π h 2 tan ⁡ θ 2 ( tan ⁡ θ 2 + sec ⁡ θ 2 ) {\displaystyle \pi h^{2}\tan {\frac {\theta }{2}}\left(\tan {\frac {\theta }{2}}+\sec {\frac {\theta }{2}}\right)} − π h 2 sin ⁡ θ 2 sin ⁡ θ 2 − 1 {\displaystyle -{\frac {\pi h^{2}\sin {\frac {\theta }{2}}}{\sin {\frac {\theta }{2}}-1}}} where θ {\displaystyle \theta } is the apex angle and h {\displaystyle h} is the height.

Circular sector

The circular sector is obtained by unfolding the surface of one nappe of the cone:

• radius R

R = r 2 + h 2 {\displaystyle R={\sqrt {r^{2}+h^{2}}}}

• arc length L

L = c = 2 π r {\displaystyle L=c=2\pi r}

• central angle φ in radians

φ = L R = 2 π r r 2 + h 2 {\displaystyle \varphi ={\frac {L}{R}}={\frac {2\pi r}{\sqrt {r^{2}+h^{2}}}}}

Equation form

The surface of a cone can be parameterized as

f ( θ , h ) = ( h cos ⁡ θ , h sin ⁡ θ , h ) , {\displaystyle f(\theta ,h)=(h\cos \theta ,h\sin \theta ,h),}

where θ ∈ [ 0 , 2 π ) {\displaystyle \theta \in [0,2\pi )} is the angle "around" the cone, and h ∈ R {\displaystyle h\in \mathbb {R} } is the "height" along the cone.

A right solid circular cone with height h {\displaystyle h} and aperture 2 θ {\displaystyle 2\theta } , whose axis is the z {\displaystyle z} coordinate axis and whose apex is the origin, is described parametrically as

F ( s , t , u ) = ( u tan ⁡ s cos ⁡ t , u tan ⁡ s sin ⁡ t , u ) {\displaystyle F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)}

where s , t , u {\displaystyle s,t,u} range over [ 0 , θ ) {\displaystyle [0,\theta )} , [ 0 , 2 π ) {\displaystyle [0,2\pi )} , and [ 0 , h ] {\displaystyle [0,h]} , respectively.

In implicit form, the same solid is defined by the inequalities

{ F ( x , y , z ) ≤ 0 , z ≥ 0 , z ≤ h } , {\displaystyle \{F(x,y,z)\leq 0,z\geq 0,z\leq h\},}

where

F ( x , y , z ) = ( x 2 + y 2 ) ( cos ⁡ θ ) 2 − z 2 ( sin ⁡ θ ) 2 . {\displaystyle F(x,y,z)=(x^{2}+y^{2})(\cos \theta )^{2}-z^{2}(\sin \theta )^{2}.\,}

More generally, a right circular cone with vertex at the origin, axis parallel to the vector d {\displaystyle d} , and aperture 2 θ {\displaystyle 2\theta } , is given by the implicit vector equation F ( u ) = 0 {\displaystyle F(u)=0} where

F ( u ) = ( u ⋅ d ) 2 − ( d ⋅ d ) ( u ⋅ u ) ( cos ⁡ θ ) 2 {\displaystyle F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \theta )^{2}} F ( u ) = u ⋅ d − | d | | u | cos ⁡ θ {\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }

where u = ( x , y , z ) {\displaystyle u=(x,y,z)} , and u ⋅ d {\displaystyle u\cdot d} denotes the dot product.

Projective geometry

In projective geometry, a cylinder is simply a cone whose apex is at infinity.¹⁰ Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as arctan, in the limit forming a right angle. This is useful in the definition of degenerate conics, which require considering the cylindrical conics.

According to G. B. Halsted, a cone is generated similarly to a Steiner conic only with a projectivity and axial pencils (not in perspective) rather than the projective ranges used for the Steiner conic:

"If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."¹¹

Generalizations

Further information: Hypercone

The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}} is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a, the vector ax is in C.¹² In this context, the analogues of circular cones are not usually special; in fact one is often interested in polyhedral cones.

An even more general concept is the topological cone, which is defined in arbitrary topological spaces.

See also

• Bicone
• Cone (linear algebra)
• Cylinder (geometry)
• Democritus
• Elliptic cone
• Generalized conic
• Hyperboloid
• List of shapes
• Pyrometric cone
• Quadric
• Rotation of axes
• Ruled surface
• Translation of axes

Notes

• Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

External links

Wikimedia Commons has media related to Cones.

• Weisstein, Eric W. "Cone". MathWorld.
• Weisstein, Eric W. "Double Cone". MathWorld.
• Weisstein, Eric W. "Generalized Cone". MathWorld.
• An interactive Spinning Cone from Maths Is Fun
• Paper model cone
• Lateral surface area of an oblique cone
• Cut a Cone An interactive demonstration of the intersection of a cone with a plane

References

1. James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410. 9780412990410 ↩

2. Grünbaum, Convex Polytopes, second edition, p. 23. /wiki/Convex_Polytopes ↩

3. Weisstein, Eric W. "Cone". MathWorld. /wiki/Eric_W._Weisstein ↩

4. James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410. 9780412990410 ↩

5. James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410. 9780412990410 ↩

6. Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01). Elementary Geometry for College Students. Cengage. ISBN 9781285965901. 9781285965901 ↩

7. Hartshorne, Robin (2013-11-11). Geometry: Euclid and Beyond. Springer Science & Business Media. Chapter 27. ISBN 9780387226767. 9780387226767 ↩

8. Blank, Brian E.; Krantz, Steven George (2006). Calculus: Single Variable. Springer. Chapter 8. ISBN 9781931914598. 9781931914598 ↩

9. Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01). Elementary Geometry for College Students. Cengage. ISBN 9781285965901. 9781285965901 ↩

10. Dowling, Linnaeus Wayland (1917-01-01). Projective Geometry. McGraw-Hill book Company, Incorporated. https://archive.org/details/projectivegeome04dowlgoog ↩

11. G. B. Halsted (1906) Synthetic Projective Geometry, page 20 /wiki/G._B._Halsted ↩

12. Grünbaum, Convex Polytopes, second edition, p. 23. /wiki/Convex_Polytopes ↩