# శంకువు

శంకువు అనేది ఒక త్రిపరిమాణ జ్యామితీయ ఆకారం. ఇది ఒక బిందువు (శీర్షం) నుండి సమతల భూమి (జ్యామితి)యొక్క అంచుల పైకి వక్రతలంగా ఉంటుంది. ఇది ఘనాకారం కలిగే త్రిపరిమాణ ఆకృతి. ఇది సమతల భూమి యొక్క చుట్టుకొలతలో గల బిందువులకు, శీర్షం నుండి కలిపే వివిధ రేఖాఖండాలతో కూడుకున్న త్రిపరిమాణమైన ఆకృతి. కొన్ని సార్లు "కోన్" అను పదం ఘనాకార పటానికి ఉపరితలంగానూ, కొన్నిసార్లు వక్రతలానికి కూడా వాడుతారు.

A right circular cone and an oblique circular cone

ఈ శంకువు యొక్క అక్షం, దాని శీర్షం గుండా పోయి ఆ ఆకారానికి భ్రమణ సౌష్టవాక్షంగా కూడా ఉంటుంది.

సాధారణంగా ప్రాథమిక జ్యామితీ భావనలలో "శంకువు" అనగా దాని భూమి వృత్తాకారంగా ఉండి దాని అక్షం ఆ వృత్తాకార భూమికి దాని శీర్షం గుండా పోతుంది. ఈ అక్షం వృత్తాకార భూమికి మధ్యలో లంబంగా కూడా ఉంటుంది. అందువల్ల దానిని "రైట్ సర్క్యులర్"గా భావిస్తారు. అనగా రైట్ (అనగా లంబంగా), సర్కులర్ (అనగా భూమి వృత్తాకారంగా) అని అర్థం. ఇలా కాకుండా "వాలు శంకువులు"లో ఈ అక్షం భూమికి లంబంగా ఉండదు.[1]

ఈ శంకువులో భూమి "బహుభుజి" ఆకారములో ఉంటే దానికి పిరమిడ్ (జ్యామితి)గా పిలుస్తారు.[2]

## ఇతర గణిత అర్థాలు

A double cone (not shown infinitely extended)

In mathematical usage, the word "cone" is used also for an "infinite cone", the union of a set of half-lines that start at a common apex point and go through a base. An infinite cone is not bounded by its base but extends to infinity. A "doubly infinite cone", or "double cone", is the union of a set of straight lines that pass through a common apex point and go through a base, therefore double infinite cones extend 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). Either half of a double cone on one side of the apex is called a "nappe".

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

## తర్వాతి పరిభాష

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 volume and the surface area for a straight cone are described in the geometry section below.

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 is called a "truncated cone"; if the truncation plane is parallel to the cone's base, it is called a 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).

## జ్యామితి

### ఉపరితల వైశాల్యం

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

${\displaystyle SA=\pi r^{2}+\pi rl}$  or
${\displaystyle SA=\pi r(r+l)}$

### ఘనపరిమాణం

The volume ${\displaystyle V}$  of any conic solid is one third of the product of the area of the base ${\displaystyle B}$  and the height ${\displaystyle H}$  (the perpendicular distance from the base to the apex).

${\displaystyle V={\frac {1}{3}}BH}$

In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ${\displaystyle \int x^{2}dx={\tfrac {1}{3}}x^{3}.}$  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.

### ద్రవ్యరాశి కేంద్రం

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.

### లంబ వృత్తాకార శంకువు

For a circular cone with radius R and height H, the formula for volume becomes

${\displaystyle V=\int _{0}^{H}r^{2}\pi \,dh}$

where r is the radius of the cone at height h measured from the apex:

${\displaystyle r=R{\frac {h}{H}}}$

Thus:

${\displaystyle V=\int _{0}^{H}\left[R{\frac {h}{H}}\right]^{2}\pi \,dh}$

Thus:

${\displaystyle V={\frac {1}{3}}\pi R^{2}H.}$

For a right circular cone, the surface area ${\displaystyle A}$  is

${\displaystyle A=\pi R^{2}+\pi RS\,}$    where   ${\displaystyle S={\sqrt {R^{2}+H^{2}}}}$    is the slant height.

The first term in the area formula, ${\displaystyle \pi R^{2}}$ , is the area of the base, while the second term, ${\displaystyle \pi RS}$ , is the area of the lateral surface.

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

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

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

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

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

where

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

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

${\displaystyle F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \theta )^{2}}$    or   ${\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }$

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

## ప్రక్షేప జ్యామితి

In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.

In projective geometry, a cylinder is simply a cone whose apex is at infinity. 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.

## ఇతర లింకులు

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