Cone

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A right circular cone and an oblique circular cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the lateral surface has a rotational symmetry.

In general, the base may be any shape, and the apex may lie anywhere (though it is often assumed that the base is bounded and has nonzero area, and that the apex lies outside the plane of the base). For example, a pyramid is technically a cone with a polygonal base. In common usage in elementary geometry, however, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.

Other mathematical meanings

In mathematical usage, the word "cone" is used also for an infinite cone, the union of any set of half-lines that start at a common apex point. This kind of cone does not have a bounding base, and extends to infinity. A doubly infinite cone, or double cone, is the union of any set of straight lines that pass through a common apex point, and therefore extends symmetrically on both sides of the apex.

The boundary of an infinite or doubly infinite cone is a conical surface, and the intersection of a plane with this surface is a conic section. For infinite cones, the word axis again usually refers to the axis of rotational symmetry (if any). 1/2 of a double cone is called a nappe.

Depending on the context, "cone" may also mean specifically a convex cone or a projective cone.

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 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.

A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum. An elliptical cone is a cone with an elliptical base. 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).

Geometry

See also: Cone (geometry) proofs

The volume $V$ of any conic solid is 1/3 the area of the base $b$ times the height $h$ (the perpendicular distance from the base to the apex).

$V = \frac{1}{3} B h$

The center of mass of a conic solid of uniform density lies 1/4 of the way from the center of mass of the base to the vertex, on the straight line joining the two.

So the volume of a cone is the volume of a cylinder divided by 3.

Right circular cone

For a circular cone with radius r and height h, the formula for volume becomes

$V=\frac{\pi}{3}r^2h$

For a right circular cone, the surface area $A$ is

$A=\pi r^2+\pi rs$ where $s = \sqrt{r^2 + h^2}$ is the slant height.

The first term in the area formula, $\pi r^2$ , is the area of the base, while the second term, $\pi r s$ , is the area of the lateral surface.

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

$S(s,t,u)=\left(u\tan(s)\cos(t),u\tan(s)\sin(t),u\right)$

where $s,t,u$ range over $[0,\theta)$ , $[0, 2\pi)$ , and $[0, h]$ , respectively.

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

$\{ S(x,y,z) \leq 0, z\geq 0, z\leq h\}$

where

$S(x,y,z)=(x^2+y^2)\cos(\theta)^2-z^2\sin(\theta)^2$

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

$S(u)=(u\cdot d)^2-(d\cdot d)(u\cdot u)\cos(\theta)^2$ or $S(u)=u\cdot d-|d||u|\cos(\theta)$

where $u=(x,y,z)$ , and $u \cdot d$ denotes the dot product.

Inscribed cone

The volume of a cone whose base is an inscribed circle is therefore:

$v=\pi\frac{4A^2h}{3P^2}$

$v=d\dfrac{\dfrac{(a^2+b^2+c^2)^2}{4}-\dfrac{a^4+b^4+c^4}{2}}{3(a+b+c)^2}\pi$

$v=\pi\frac{n^2}{3(6n-12)^2\tan^2\left(\frac{180}{n}\right)}s^3$

$v=\pi\frac{\sin^2\left(\frac{360}{n}\right)}{12\left(1+\cos\left(\frac{180}{n}\right)+\sin\left(\frac{180}{n}\right)\right)^2}s^3$

$v=\frac{\pi}{12\tan^2\left(\frac{180}{n}\right)}s^3$

The surface area of a cone whose base is an inscribed circle is therefore:

$sa=\pi\left(\frac{4A^2}{P^2}+\sqrt{\frac{16A^4}{P^4}+\frac{4h^2A^2}{P^2}}\right)$

$sa= \pi\frac{\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2})}{(a+b+c)^2} + \sqrt{d^2\frac{\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2}}{(a+b+c)^2} + \frac{(\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2})^2}{(a+b+c)^4}}$

$sa=s^2\pi(\frac{n^2}{(6n-12)^2\tan^2(\frac{180}{n})}+\sqrt{\frac{n^2}{(6n-12)^2\tan^2(\frac{180}{n})}+\frac{n^4}{(6n-12)^4\tan^4(\frac{180}{n})}})$

$sa= s^2\pi(\frac{sin^2(\frac{360}{n})}{4(1+cos(\frac{180}{n})+sin(\frac{180}{n}))^2} + \sqrt{\frac{sin^4(\frac{360}{n})}{16(1+sin(\frac{180}{n})+cos(\frac{180}{n}))^4} + \frac{sin^2(\frac{180}{n})}{4(1+\cos\left(\frac{180}{n}\right)+\sin\left(\frac{180}{n}\right))^2}})$

$sa=\pi\left(\frac{1}{4\tan^2\left(\frac{180}{n}\right)}+\sqrt{\frac{1}{4\tan^2\left(\frac{180}{n}\right)}+\frac{1}{16tan^4(\frac{180}{n})}}\right)s^2$

Circumscribed cone

The volume of a cone whose base is an circumscribed circle is therefore:

$V= \pi\frac{a^2b^2c^2h}{48A^2}$

$V= \frac{a^2b^2c^2d}{3(a^2+b^2+c^2)^2-6(a^4 + b^4 +c^4)}\pi$

$V= s^3\frac{\pi}{12}$

$V= s^3\pi\frac{(1-\frac{2}{n})^2}{3}tan^2(\frac{180}{n})$

$V= s^3 \frac{\pi}{12sin^2(\frac{180}{n})}$

The surface area of a cone whose base is an circumscribed circle is therefore:

$SA= \pi(\frac{a^2b^2c^2}{16A^2} + \sqrt{\frac{a^2b^2c^2h^2}{16A^2} + \frac{a^4b^4c^4}{256A^4}})$

$SA= \pi(\frac{a^2b^2c^2}{(a^2+b^2+c^2)^2-2(a^4 + b^4 +c^4)} + \sqrt{\frac{a^2 b^2 c^2 d^2}{(a^2+b^2+c^2)^2- 2(a^4+b^4+c^4)} + \frac{a^4b^4c^4}{((a^2+b^2+c^2)^2- 2(a^4+b^4+c^4))^2}})$

$SA= s^2\pi(\frac{1}{4} + \sqrt{\frac{5}{16}})$

$SA= s^2\pi((1-\frac{2}{n})^2tan^2(\frac{180}{n}) + \sqrt{(1-\frac{2}{n})^2tan^2(\frac{180}{n}) + (1-\frac{2}{n})^4tan^4(\frac{180}{n})})$

$SA= s^2\pi(\frac{1}{4sin^2(\frac{180}{n})} + \sqrt{\frac{1}{4sin^2(\frac{180}{n})} + \frac{1}{16sin^4(\frac{180}{n})}})$

Bicone

A bicone or dicone is a solid formed by joining an cone and its mirror image base-to-base.

the name of the bicones is not an external face but an internal one, existing on the primary symmetry plane which connects the two cone halves.

The face-transitive cones are the dual polyhedra of the uniform cylinder and will generally have round faces.

A bicone can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator.

For a circular bicone with radius r and height h, the formula for volume becomes

$V = \frac{2}{3} \pi r^2 h.$

Surface area of a bicone with diameter $d = 2r$, slant height $L= \sqrt{r^2 + h^2}$ is:

$SA= dL\pi$

Inscribed bicone

Volume

The volume of a bicone whose base is an inscribed circle is therefore:

$v= h\pi\frac{8A^2}{3P^2}$

$v= d\frac{\frac{(a^2+b^2+c^2)^2}{2}-(a^4+b^4+c^4)}{3(a+b+c)^2}\pi$

$v= s^3\pi\frac{2n^2}{3(6n-12)^2tan^2(\frac{180}{n})}$

$v= s^3\pi\frac{sin^2(\frac{360}{n})}{6(1+cos(\frac{180}{n})+sin(\frac{180}{n}))^2}$

$v= s^3 \frac{\pi}{6tan^2(\frac{180}{n})}$

Surface area

The surface area of a bicone whose base is an inscribed circle is therefore:

$sa= \pi\sqrt{\frac{64A^4}{P^4} + \frac{16h^2A^2}{P^2}}$

$sa= 2\pi\sqrt{d^2\frac{\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2}}{(a+b+c)^2} + \frac{(\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2})^2}{(a+b+c)^4}}$

$sa= s^2\pi\sqrt{\frac{(2n)^2}{(6n-12)^2tan^2(\frac{180}{n})} + \frac{4n^4}{(6n-12)^4tan^4(\frac{180}{n})}}$

$sa= s^2\pi\sqrt{\frac{sin^4(\frac{360}{n})}{4(1+sin(\frac{180}{n})+cos(\frac{180}{n}))^4} + \frac{sin^2(\frac{180}{n})}{(1+cos(\frac{180}{n})+sin(\frac{180}{n}))^2}}$

$sa= s^2\pi\sqrt{\frac{4}{4tan^2(\frac{180}{n})} + \frac{1}{4tan^4(\frac{180}{n})}}$

Circumscribed bicone

Volume

The volume of a bicone whose base is an circumscribed circle is therefore:

$V= \pi\frac{a^2b^2c^2h}{24A^2}$

$V= \frac{2a^2b^2c^2d}{3(a^2+b^2+c^2)^2-6(a^4 + b^4 +c^4)}\pi$

$V= s^3\frac{\pi}{6}$

$V= s^3\pi\frac{2(1-\frac{2}{n})^2}{3}tan^2(\frac{180}{n})$

$V= s^3 \frac{\pi}{6sin^2(\frac{180}{n})}$

Surface area

The surface area of a bicone whose base is an circumscribed circle is therefore:

$SA= \pi \sqrt{\frac{a^2b^2c^2h^2}{(2A)^2} + \frac{a^4b^4c^4}{64A^4}}$

$SA= 2\pi\sqrt{\frac{a^2 b^2 c^2 d^2}{(a^2+b^2+c^2)^2- 2(a^4+b^4+c^4)}+ \frac{a^4b^4c^4}{((a^2+b^2+c^2)^2- 2(a^4+b^4+c^4))^2}}$

$SA= s^2\pi\sqrt{\frac{5}{4}}$

$SA= s^22\pi\sqrt{(1-\frac{2}{n})^2tan^2(\frac{180}{n})+ (1-\frac{2}{n})^4tan^4(\frac{180}{n})}$

$SA= s^2\pi\sqrt{\frac{4}{4sin^2(\frac{180}{n})} + \frac{1}{4sin^4(\frac{180}{n})}}$

Derivation of cone formulae from pyramid formulae

Lateral Area

The lateral area of a cone may be derived by approximating it with an n-sided pyramid and taking the limit as n approaches infinity.

Let

$\beta = \frac{180^{\circ}}{n} = \frac{\pi}{n}$

where n is the number of sides of the pyramid's base. The pyramid's lateral area is given by

$A_n = nR \sin \beta \sqrt{R^2 \cos^2 \beta + h^2}$

Taking the limit as n approaches infinity, we obtain the lateral area of the cone:

$A_L = \lim_{n \to \infty} nR \sin \beta \sqrt{R^2 \cos^2 \beta + h^2}$

Since R is constant, this can be rewritten as:

$R \left( \lim_{n \to \infty} n \sin \beta \right) \left( \lim_{n \to \infty} \sqrt{R^2 \cos^2 \beta + h^2} \right)$

Each limit in this expression can be evaluated separately.

1. $\lim_{n \to \infty} n \sin \beta$

This expression can be proved to be equal to $\pi$. Let L be the length of an edge in the pyramid's polygonal base. The relation between L and the polygon's radius R is

$L = 2R \sin \beta$

The perimeter of the polygon is given by:

$P = n \cdot L$

Therefore, the perimeter of a regular polygon is:

$P = 2nR \sin \beta$

Taking the limit as n approaches infinity, we obtain the perimeter of the circular base of the cone:

$P_\circ = \lim_{n \to \infty} 2nR \sin \beta$

But since the ratio of a circle's perimeter to its diameter 2R is equal to $\pi$, we have:

$\pi = \frac{\lim_{n \to \infty} 2nR \sin \beta}{2R} = \lim_{n \to \infty} \frac{2nR \sin \beta}{2R}$

Since 2R is constant and non-zero, this expression can be divided by 2R both in the nominator and the denominator, giving:

$\pi = \lim_{n \to \infty} n \sin \beta$

2. $\lim_{n \to \infty} \sqrt{R^2 \cos^2 \beta + h^2}$

Since $\lim_{n \to \infty} \frac{\pi}{n} = 0$, we have:

$\lim_{n \to \infty} \cos^2 \frac{\pi}{n} = \cos^2 0 = 1^2= 1.$

So, the original square root becomes

$\sqrt{R^2 + h^2}$

This expression is equal to the hypotenuse $\delta$ of the right triangle whose height is the height of the cone, and whose base is the radius of the cone. Therefore, $\delta$ is equal to the length of the line from the apex of the cone to its circular edge.

By putting together the elements of the original equation, we obtain:

$R \left( \lim_{n \to \infty} n \sin \beta \right) \left( \lim_{n \to \infty} \sqrt{R^2 \cos^2 \beta + h^2} \right) = \pi R \delta$

Therefore, the lateral area $A_L$ of a cone is given by

$A_L=\pi R \delta$.

See also

• Cone (topology)
• Democritus
• Pyramid (geometry)
• Conic section
• Quadric
• Ruled surface
• Hyperboloid
• Ice cream cone

External links

• Spinning Cone from Math Is Fun
• Paper model cone
• Lateral surface area of an oblique cone
• Generalized Cone from Wolfram MathWorld