Cone
Cone | |
---|---|
Type | Solid figure |
Faces | 1 circular face and 1 conic surface |
Euler char. | 2 |
Symmetry group | O(2) |
Surface area | πr2 + πrℓ |
Volume | (πr2h)/3 |
A cone is a
A cone is formed by a set of
In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe.
The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry.
In common usage in elementary
A cone with a polygonal base is called a pyramid.
Depending on the context, "cone" may also mean specifically a convex cone or a projective cone.
Cones can also be generalized to higher dimensions.
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
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.[1] An elliptical cone is a cone with an elliptical base.[1] A generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary (also see visual hull).
Measurements and equations
Volume
The volume of any conic solid is one third of the product of the area of the base and the height [4]
In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral
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 and height h, the base is a circle of area and so the formula for volume becomes[6]
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 , where is the radius of the base and is the height. This can be proved by the Pythagorean theorem.
Surface area
The lateral surface area of a right circular cone is where is the radius of the circle at the bottom of the cone and is the slant height of the cone.[4] The surface area of the bottom circle of a cone is the same as for any circle, . Thus, the total surface area of a right circular cone can be expressed as each of the following:
- Radius and height
- (the area of the base plus the area of the lateral surface; the term is the slant height)
- where is the radius and is the height.
- Radius and slant height
- where is the radius and is the slant height.
- Circumference and slant height
- where is the circumference and is the slant height.
- Apex angle and height
- where is the apex angle and is the height.
Circular sector
The circular sector is obtained by unfolding the surface of one nappe of the cone:
- radius R
- arc length L
- central angle φ in radians
Equation form
The surface of a cone can be parameterized as
where is the angle "around" the cone, and is the "height" along the cone.
A right solid circular cone with height and aperture , whose axis is the coordinate axis and whose apex is the origin, is described parametrically as
where range over , , and , respectively.
In implicit form, the same solid is defined by the inequalities
where
More generally, a right circular cone with vertex at the origin, axis parallel to the vector , and aperture , is given by the implicit vector equation where
where , and denotes the dot product.
Elliptic cone
In the Cartesian coordinate system, an elliptic cone is the locus of an equation of the form[7]
It is an
- Any plane section of an elliptic cone is a conic section.
Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see circular section).
The intersection of an elliptic cone with a concentric sphere is a spherical conic.
Projective geometry
In
According to
"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'."[9]
Generalizations
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 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.[2] 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
See also
- Bicone
- Cone (linear algebra)
- Cylinder (geometry)
- Democritus
- Generalized conic
- Hyperboloid
- List of shapes
- Pyrometric cone
- Quadric
- Rotation of axes
- Ruled surface
- Translation of axes
Notes
- ^ ISBN 9780412990410.
- ^ a b Grünbaum, Convex Polytopes, second edition, p. 23.
- ^ Weisstein, Eric W. "Cone". MathWorld.
- ^ ISBN 9781285965901.
- ISBN 9780387226767.
- ISBN 9781931914598.
- ^ Protter & Morrey (1970, p. 583)
- ^ Dowling, Linnaeus Wayland (1917-01-01). Projective Geometry. McGraw-Hill book Company, Incorporated.
- ^ G. B. Halsted (1906) Synthetic Projective Geometry, page 20
References
- Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: LCCN 76087042
External links
- 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