## Friday, 13 September 2013

### Cone its Area and Volume

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base, such that there is a circular cross section. 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 base has a rotational symmetry.
In common usage in elementary geometry, 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. In general, however, the base may be any shape that permits a circular cross section of the cone, and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base).
The more general conic solid also has an an apex and lines connecting the apex to all the points on a planar base, which can be of any shape. If the result has a circular cross section, it is a cone; if it has a polygonal base, it is a pyramid.

### Surface area

The lateral surface area of a right circular cone is $LSA = \pi r l$ where $r$ is the radius of the circle at the bottom of the cone and $l$ is the lateral height of the cone (given by the Pythagorean theorem $l=\sqrt{r^2+h^2}$ where $h$ is the height of the cone). The surface area of the bottom circle of a cone is the same as for any circle, $\pi r^2$. Thus the total surface area of a right circular cone is:
$SA=\pi r^2+\pi r l$ or
$SA=\pi r(r+l)$

### Volume

The volume $V$ of any conic solid is one third of the product of the area of the base $B$ and the height $H$ (the perpendicular distance from the base to the apex).
$V = \frac{1}{3} B H$
In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral $\int x^2 dx = \tfrac{1}{3} x^3.$ 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.

### 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

For a circular cone with radius R and height H, the formula for volume becomes
$V = \int_0^H r^2 \pi \, dh$
where r is the radius of the cone at height h measured from the apex:
$r= R \frac{h}{H}$
Thus:
$V = \int_0^H \left[R \frac{h}{H}\right]^2 \pi \, dh$
Thus:
$V = \frac{1}{3} \pi R^2 H.$
For a right circular cone, the surface area $A$ is
$A =\pi R^2 + \pi R S\,$   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
$F(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)$
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
$\{ F(x,y,z) \leq 0, z\geq 0, z\leq h\},$
where
$F(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.\,$
More generally, a right circular cone with vertex at the origin, axis parallel to the vector $d$, and aperture $2\theta$, is given by the implicit vector equation $F(u) = 0$ where
$F(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2$   or   $F(u) = u \cdot d - |d| |u| \cos \theta$
where $u=(x,y,z)$, and $u \cdot d$ denotes the dot product.