418:
54:
329:
3329:
3117:
1279:
680:– 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
184:
193:
176:
370:
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
270:
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
1709:
1789:
2752:
1563:
2054:
2949:
3221:
2393:
2147:
1490:
1228:
3352:
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
1108:
2612:
674:
3029:
808:
1623:
1909:
609:
325:, and that the apex lies outside the plane of the base). Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly.
1156:
864:
3291:
2191:
1333:
2219:
3444:
1378:
962:
3386:"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'."
2460:
1962:
3076:
1032:
753:
3102:
2798:
2265:
1813:
2837:
2428:
1607:
1422:
1002:
538:
2775:
2285:
2242:
1833:
1715:
1587:
1402:
1272:
1252:
982:
908:
884:
558:
511:
2522:
2495:
321:. In general, however, the base may be any shape and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite
2623:
1502:
1983:
692:(can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.
704:
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.
2845:
434:
3340:
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.
3142:
2293:
617:
417:
3710:
3683:
3653:
3580:
259:
object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the
108:
2072:
1438:
1167:
3328:
1047:
53:
2537:
328:
3125:
2955:
3891:
3309:
Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see
685:
286:
of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a
251:, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a
1704:{\displaystyle \pi h^{2}\tan {\frac {\theta }{2}}\left(\tan {\frac {\theta }{2}}+\sec {\frac {\theta }{2}}\right)}
1004:
is the slant height of the cone. The surface area of the bottom circle of a cone is the same as for any circle,
761:
372:
3466:. In this context, the analogues of circular cones are not usually special; in fact one is often interested in
614:
In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral
677:
1864:
565:
392:
of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle
473:
is the surface created by the set of lines passing through a vertex and every point on a boundary (also see
206:
1117:
825:
212:
that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the
3237:
2155:
1296:
3491:
448:
248:
31:
3365:
2196:
3875:
3420:
317:
to its plane. If the cone is right circular the intersection of a plane with the lateral surface is a
3380:
681:
98:
1345:
1034:. Thus, the total surface area of a right circular cone can be expressed as each of the following:
929:
3541:
3496:
3349:
3345:
3337:
3333:
3133:
911:
283:
676:
Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying
2433:
341:
1929:
3037:
2465:
1007:
728:
3896:
3841:
3822:
3803:
3784:
3706:
3679:
3649:
3615:
3576:
3506:
3081:
2780:
2528:
2247:
287:
217:
2757:
More generally, a right circular cone with vertex at the origin, axis parallel to the vector
1798:
3669:
3600:
3531:
3474:
3361:
3310:
2807:
376:
239:
that does not contain the apex. Depending on the author, the base may be restricted to be a
236:
113:
2401:
1592:
1407:
987:
516:
3566:
3521:
3317:
2801:
1845:
1784:{\displaystyle -{\frac {\pi h^{2}\sin {\frac {\theta }{2}}}{\sin {\frac {\theta }{2}}-1}}}
923:
426:
352:
322:
264:
256:
232:
213:
209:
88:
38:
822:
of its base to the apex via a line segment along the surface of the cone. It is given by
3739:
3116:
3780:
3516:
2760:
2270:
2227:
1818:
1572:
1387:
1257:
1237:
967:
893:
869:
701:
543:
496:
252:
244:
228:
3870:
3865:
17:
3885:
3844:
3754:
3536:
3376:
3372:
3294:
318:
314:
3415:
3383:(not in perspective) rather than the projective ranges used for the Steiner conic:
2747:{\displaystyle F(x,y,z)=(x^{2}+y^{2})(\cos \theta )^{2}-z^{2}(\sin \theta )^{2}.\,}
433:
224:
122:
3700:
3673:
3643:
3570:
235:
connecting a common point, the apex, to all of the points on a base that is in a
3825:
3511:
3467:
3412:
3401:
3357:
3105:
1558:{\displaystyle \left({\frac {c}{2}}\right)\left({\frac {c}{2\pi }}+\ell \right)}
1278:
818:
The slant height of a right circular cone is the distance from any point on the
474:
348:
3859:
1282:
Total surface area of a right circular cone, given radius 𝑟 and slant height ℓ
3501:
3405:
3227:
2287:
coordinate axis and whose apex is the origin, is described parametrically as
2049:{\displaystyle \varphi ={\frac {L}{R}}={\frac {2\pi r}{\sqrt {r^{2}+h^{2}}}}}
3849:
3830:
3811:
3623:
3395:
359:
3806:
3618:
2944:{\displaystyle F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \theta )^{2}}
294:
3878:
An interactive demonstration of the intersection of a cone with a plane
3526:
3297:
is a conic section of the same type (ellipse, parabola,...), one gets:
513:
of any conic solid is one third of the product of the area of the base
464:
453:
441:
A cone with a region including its apex cut off by a plane is called a
337:
183:
3486:
3353:
1114:(the area of the base plus the area of the lateral surface; the term
887:
819:
491:
401:
386:
of its base; often this is simply called the radius of the cone. The
383:
306:
240:
150:
3216:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=z^{2}.}
3400:
The definition of a cone may be extended to higher dimensions; see
3316:
The intersection of an elliptic cone with a concentric sphere is a
2388:{\displaystyle F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)}
275:. Either half of a double cone on one side of the apex is called a
192:
175:
3327:
3115:
1277:
432:
416:
332:
Air traffic control tower in the shape of a cone, Sharjah
Airport.
327:
190:
182:
174:
3788:
1848:
is obtained by unfolding the surface of one nappe of the cone:
347:
Depending on the context, "cone" may also mean specifically a
2142:{\displaystyle f(\theta ,h)=(h\cos \theta ,h\sin \theta ,h),}
3642:
Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01).
157:
137:
128:
1485:{\displaystyle {\frac {c^{2}}{4\pi }}+{\frac {c\ell }{2}}}
984:
is the radius of the circle at the bottom of the cone and
313:
means that the axis passes through the centre of the base
3575:. Springer Science & Business Media. pp. 74–75.
1223:{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}
3446:
is a cone (with apex at the origin) if for every vector
645:
3699:
Blank, Brian E.; Krantz, Steven George (2006-01-01).
3678:. Springer Science & Business Media. Chapter 27.
3423:
3240:
3145:
3084:
3040:
2958:
2848:
2810:
2783:
2763:
2626:
2540:
2498:
2468:
2436:
2404:
2296:
2273:
2250:
2230:
2199:
2158:
2075:
1986:
1932:
1867:
1821:
1801:
1718:
1626:
1595:
1575:
1505:
1441:
1410:
1390:
1348:
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1240:
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1120:
1050:
1010:
990:
970:
932:
896:
872:
828:
764:
731:
620:
568:
546:
519:
499:
451:
plane is parallel to the cone's base, it is called a
3775:
Protter, Murray H.; Morrey, Charles B. Jr. (1970),
3705:. Springer Science & Business Media. Chapter 8.
3477:, which is defined in arbitrary topological spaces.
2531:
form, the same solid is defined by the inequalities
1103:{\displaystyle \pi r^{2}+\pi r{\sqrt {r^{2}+h^{2}}}}
2607:{\displaystyle \{F(x,y,z)\leq 0,z\geq 0,z\leq h\},}
669:{\displaystyle \int x^{2}\,dx={\tfrac {1}{3}}x^{3}}
149:
121:
107:
97:
87:
79:
46:
3438:
3285:
3215:
3096:
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3023:
2943:
2831:
2792:
2769:
2746:
2606:
2516:
2489:
2454:
2422:
2387:
2279:
2259:
2236:
2213:
2185:
2141:
2048:
1956:
1903:
1827:
1807:
1783:
1703:
1601:
1581:
1557:
1484:
1416:
1396:
1372:
1327:
1266:
1246:
1222:
1150:
1102:
1026:
996:
976:
956:
902:
878:
858:
802:
747:
688:– more precisely, not all polyhedral pyramids are
668:
603:
552:
532:
505:
414:of the cone, to distinguish it from the aperture.
179:A right circular cone and an oblique circular cone
58:A right circular cone with the radius of its base
3024:{\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }
263:; if the lateral surface is unbounded, it is a
2066:The surface of a cone can be parameterized as
3725:
187:A double cone (not shown infinitely extended)
8:
2598:
2541:
382:The "base radius" of a circular cone is the
803:{\displaystyle V={\frac {1}{3}}\pi r^{2}h.}
3594:
3592:
3293:From the fact, that the affine image of a
52:
3744:. McGraw-Hill book Company, Incorporated.
3430:
3426:
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3422:
3271:
3258:
3245:
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3179:
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3083:
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2782:
2762:
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2024:
2006:
1993:
1985:
1931:
1893:
1880:
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1820:
1800:
1762:
1744:
1732:
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1717:
1686:
1667:
1646:
1634:
1625:
1594:
1574:
1529:
1510:
1504:
1467:
1448:
1442:
1440:
1409:
1389:
1347:
1307:
1298:
1259:
1239:
1207:
1194:
1188:
1169:
1140:
1127:
1121:
1119:
1092:
1079:
1073:
1058:
1049:
1018:
1009:
989:
969:
931:
910:is the height. This can be proved by the
895:
871:
848:
835:
829:
827:
788:
771:
763:
739:
730:
660:
644:
634:
628:
619:
589:
575:
567:
545:
524:
518:
498:
3738:Dowling, Linnaeus Wayland (1917-01-01).
3645:Elementary Geometry for College Students
2224:A right solid circular cone with height
3871:Lateral surface area of an oblique cone
3777:College Calculus with Analytic Geometry
3553:
3305:of an elliptic cone is a conic section.
1904:{\displaystyle R={\sqrt {r^{2}+h^{2}}}}
604:{\displaystyle V={\frac {1}{3}}A_{B}h.}
3360:. This is useful in the definition of
755:and so the formula for volume becomes
43:
3375:, a cone is generated similarly to a
1151:{\displaystyle {\sqrt {r^{2}+h^{2}}}}
859:{\displaystyle {\sqrt {r^{2}+h^{2}}}}
684:. This is essentially the content of
437:A cone truncated by an inclined plane
7:
3637:
3635:
3561:
3559:
3557:
3473:An even more general concept is the
3286:{\displaystyle x^{2}+y^{2}=z^{2}\ .}
2193:is the angle "around" the cone, and
2186:{\displaystyle \theta \in [0,2\pi )}
1328:{\displaystyle \pi r^{2}+\pi r\ell }
93:1 circular face and 1 conic surface
3454:and every nonnegative real number
3120:An elliptical cone quadric surface
25:
2214:{\displaystyle h\in \mathbb {R} }
926:area of a right circular cone is
358:Cones can also be generalized to
3439:{\displaystyle \mathbb {R} ^{n}}
3404:. In this case, one says that a
3364:, which require considering the
2221:is the "height" along the cone.
717:For a circular cone with radius
725:, the base is a circle of area
3065:
3047:
3008:
3000:
2995:
2987:
2968:
2962:
2932:
2919:
2916:
2904:
2901:
2889:
2877:
2864:
2858:
2852:
2820:
2814:
2731:
2718:
2696:
2683:
2680:
2654:
2648:
2630:
2565:
2547:
2511:
2499:
2484:
2469:
2449:
2437:
2318:
2300:
2180:
2165:
2133:
2097:
2091:
2079:
1429:Circumference and slant height
1373:{\displaystyle \pi r(r+\ell )}
1367:
1355:
957:{\displaystyle LSA=\pi r\ell }
396:to the axis, the aperture is 2
293:In common usage in elementary
1:
3759:Synthetic Projective Geometry
3569:; James, Glenn (1992-07-31).
3379:only with a projectivity and
223:A cone is formed by a set of
3675:Geometry: Euclid and Beyond
3136:of an equation of the form
3126:Cartesian coordinate system
2800:, is given by the implicit
2455:{\displaystyle [0,\theta )}
3913:
3726:Protter & Morrey (1970
3572:The Mathematics Dictionary
3393:
1957:{\displaystyle L=c=2\pi r}
481:Measurements and equations
297:, cones are assumed to be
36:
29:
3779:(2nd ed.), Reading:
3702:Calculus: Single Variable
3356:, in the limit forming a
3071:{\displaystyle u=(x,y,z)}
2490:{\displaystyle [0,2\pi )}
1589:is the circumference and
1027:{\displaystyle \pi r^{2}}
748:{\displaystyle \pi r^{2}}
423:Problemata mathematica...
305:means that the base is a
247:in the plane, any closed
51:
3604:, second edition, p. 23.
3097:{\displaystyle u\cdot d}
2793:{\displaystyle 2\theta }
2260:{\displaystyle 2\theta }
375:of a conic section, see
37:Not to be confused with
1808:{\displaystyle \theta }
1287:Radius and slant height
686:Hilbert's third problem
3440:
3341:
3287:
3230:of the right-circular
3217:
3121:
3098:
3072:
3025:
2945:
2833:
2832:{\displaystyle F(u)=0}
2794:
2771:
2748:
2608:
2518:
2491:
2456:
2424:
2389:
2281:
2261:
2238:
2215:
2187:
2143:
2050:
1958:
1905:
1829:
1815:is the apex angle and
1809:
1785:
1705:
1603:
1583:
1559:
1486:
1418:
1398:
1374:
1329:
1283:
1268:
1248:
1224:
1152:
1104:
1028:
998:
978:
958:
904:
880:
860:
804:
749:
670:
605:
554:
534:
507:
438:
430:
333:
249:one-dimensional figure
243:, any one-dimensional
198:
188:
180:
18:Cone (geometry)/Proofs
3492:Cone (linear algebra)
3441:
3394:Further information:
3331:
3288:
3218:
3119:
3099:
3073:
3026:
2946:
2834:
2795:
2772:
2749:
2609:
2519:
2492:
2457:
2425:
2423:{\displaystyle s,t,u}
2390:
2282:
2262:
2239:
2216:
2188:
2144:
2051:
1959:
1906:
1830:
1810:
1786:
1706:
1614:Apex angle and height
1604:
1602:{\displaystyle \ell }
1584:
1560:
1487:
1419:
1417:{\displaystyle \ell }
1399:
1375:
1330:
1281:
1269:
1249:
1225:
1153:
1105:
1029:
999:
997:{\displaystyle \ell }
979:
959:
905:
881:
861:
805:
750:
678:Cavalieri's principle
671:
606:
555:
535:
533:{\displaystyle A_{B}}
508:
436:
420:
331:
196:
186:
178:
32:Cone (disambiguation)
3648:. Cengage Learning.
3421:
3238:
3143:
3082:
3038:
2956:
2846:
2808:
2781:
2761:
2624:
2538:
2496:
2466:
2434:
2402:
2294:
2271:
2267:, whose axis is the
2248:
2228:
2197:
2156:
2073:
1984:
1930:
1865:
1819:
1799:
1716:
1624:
1609:is the slant height.
1593:
1573:
1503:
1439:
1424:is the slant height.
1408:
1388:
1346:
1297:
1258:
1238:
1168:
1158:is the slant height)
1118:
1048:
1008:
988:
968:
930:
894:
870:
826:
762:
729:
682:method of exhaustion
618:
566:
544:
517:
497:
366:Further terminology
255:; otherwise it is a
30:For other uses, see
3741:Projective Geometry
3542:Translation of axes
3497:Cylinder (geometry)
3346:projective geometry
3334:projective geometry
3324:Projective geometry
912:Pythagorean theorem
708:Right circular cone
66:, its slant height
3845:"Generalized Cone"
3842:Weisstein, Eric W.
3823:Weisstein, Eric W.
3804:Weisstein, Eric W.
3616:Weisstein, Eric W.
3436:
3366:cylindrical conics
3342:
3283:
3213:
3122:
3094:
3068:
3021:
2941:
2829:
2790:
2767:
2744:
2604:
2514:
2487:
2452:
2420:
2385:
2277:
2257:
2234:
2211:
2183:
2139:
2046:
1954:
1901:
1825:
1805:
1781:
1701:
1599:
1579:
1555:
1482:
1414:
1404:is the radius and
1394:
1370:
1325:
1284:
1264:
1254:is the radius and
1244:
1220:
1148:
1100:
1024:
994:
974:
954:
900:
876:
856:
800:
745:
690:scissors congruent
666:
654:
601:
550:
530:
503:
463:is a cone with an
439:
431:
421:Illustration from
334:
199:
197:3D model of a cone
189:
181:
3892:Elementary shapes
3862:from Maths Is Fun
3670:Hartshorne, Robin
3507:Generalized conic
3362:degenerate conics
3279:
3195:
3168:
2770:{\displaystyle d}
2280:{\displaystyle z}
2237:{\displaystyle h}
2044:
2043:
2001:
1899:
1828:{\displaystyle h}
1779:
1770:
1752:
1694:
1675:
1654:
1582:{\displaystyle c}
1542:
1518:
1480:
1462:
1397:{\displaystyle r}
1267:{\displaystyle h}
1247:{\displaystyle r}
1213:
1146:
1098:
1038:Radius and height
977:{\displaystyle r}
903:{\displaystyle h}
879:{\displaystyle r}
854:
779:
653:
583:
553:{\displaystyle h}
506:{\displaystyle V}
360:higher dimensions
340:base is called a
288:circular symmetry
207:three-dimensional
173:
172:
16:(Redirected from
3904:
3866:Paper model cone
3855:
3854:
3836:
3835:
3817:
3816:
3791:
3762:
3752:
3746:
3745:
3735:
3729:
3723:
3717:
3716:
3696:
3690:
3689:
3666:
3660:
3659:
3639:
3630:
3629:
3628:
3611:
3605:
3601:Convex Polytopes
3596:
3587:
3586:
3563:
3532:Rotation of axes
3475:topological cone
3468:polyhedral cones
3445:
3443:
3442:
3437:
3435:
3434:
3429:
3311:circular section
3292:
3290:
3289:
3284:
3277:
3276:
3275:
3263:
3262:
3250:
3249:
3222:
3220:
3219:
3214:
3209:
3208:
3196:
3194:
3193:
3184:
3183:
3174:
3169:
3167:
3166:
3157:
3156:
3147:
3103:
3101:
3100:
3095:
3077:
3075:
3074:
3069:
3030:
3028:
3027:
3022:
3011:
3003:
2998:
2990:
2950:
2948:
2947:
2942:
2940:
2939:
2885:
2884:
2838:
2836:
2835:
2830:
2799:
2797:
2796:
2791:
2776:
2774:
2773:
2768:
2753:
2751:
2750:
2745:
2739:
2738:
2717:
2716:
2704:
2703:
2679:
2678:
2666:
2665:
2613:
2611:
2610:
2605:
2524:, respectively.
2523:
2521:
2520:
2517:{\displaystyle }
2515:
2494:
2493:
2488:
2461:
2459:
2458:
2453:
2429:
2427:
2426:
2421:
2394:
2392:
2391:
2386:
2384:
2380:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2243:
2241:
2240:
2235:
2220:
2218:
2217:
2212:
2210:
2192:
2190:
2189:
2184:
2148:
2146:
2145:
2140:
2055:
2053:
2052:
2047:
2045:
2042:
2041:
2029:
2028:
2019:
2018:
2007:
2002:
1994:
1963:
1961:
1960:
1955:
1910:
1908:
1907:
1902:
1900:
1898:
1897:
1885:
1884:
1875:
1834:
1832:
1831:
1826:
1814:
1812:
1811:
1806:
1790:
1788:
1787:
1782:
1780:
1778:
1771:
1763:
1754:
1753:
1745:
1737:
1736:
1723:
1710:
1708:
1707:
1702:
1700:
1696:
1695:
1687:
1676:
1668:
1655:
1647:
1639:
1638:
1608:
1606:
1605:
1600:
1588:
1586:
1585:
1580:
1564:
1562:
1561:
1556:
1554:
1550:
1543:
1541:
1530:
1523:
1519:
1511:
1491:
1489:
1488:
1483:
1481:
1476:
1468:
1463:
1461:
1453:
1452:
1443:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1379:
1377:
1376:
1371:
1334:
1332:
1331:
1326:
1312:
1311:
1273:
1271:
1270:
1265:
1253:
1251:
1250:
1245:
1229:
1227:
1226:
1221:
1219:
1215:
1214:
1212:
1211:
1199:
1198:
1189:
1157:
1155:
1154:
1149:
1147:
1145:
1144:
1132:
1131:
1122:
1109:
1107:
1106:
1101:
1099:
1097:
1096:
1084:
1083:
1074:
1063:
1062:
1033:
1031:
1030:
1025:
1023:
1022:
1003:
1001:
1000:
995:
983:
981:
980:
975:
963:
961:
960:
955:
909:
907:
906:
901:
890:of the base and
885:
883:
882:
877:
865:
863:
862:
857:
855:
853:
852:
840:
839:
830:
809:
807:
806:
801:
793:
792:
780:
772:
754:
752:
751:
746:
744:
743:
675:
673:
672:
667:
665:
664:
655:
646:
633:
632:
610:
608:
607:
602:
594:
593:
584:
576:
559:
557:
556:
551:
539:
537:
536:
531:
529:
528:
512:
510:
509:
504:
470:generalized cone
377:Dandelin spheres
195:
169:
160:
145:
140:
131:
116:
56:
44:
21:
3912:
3911:
3907:
3906:
3905:
3903:
3902:
3901:
3882:
3881:
3858:An interactive
3840:
3839:
3821:
3820:
3802:
3801:
3798:
3774:
3771:
3766:
3765:
3753:
3749:
3737:
3736:
3732:
3724:
3720:
3713:
3698:
3697:
3693:
3686:
3668:
3667:
3663:
3656:
3641:
3640:
3633:
3614:
3613:
3612:
3608:
3597:
3590:
3583:
3565:
3564:
3555:
3550:
3522:Pyrometric cone
3483:
3424:
3419:
3418:
3398:
3392:
3390:Generalizations
3326:
3318:spherical conic
3267:
3254:
3241:
3236:
3235:
3200:
3185:
3175:
3158:
3148:
3141:
3140:
3114:
3080:
3079:
3036:
3035:
2954:
2953:
2931:
2876:
2844:
2843:
2806:
2805:
2779:
2778:
2777:, and aperture
2759:
2758:
2730:
2708:
2695:
2670:
2657:
2622:
2621:
2536:
2535:
2464:
2463:
2432:
2431:
2400:
2399:
2328:
2324:
2292:
2291:
2269:
2268:
2246:
2245:
2226:
2225:
2195:
2194:
2154:
2153:
2071:
2070:
2064:
2033:
2020:
2008:
1982:
1981:
1928:
1927:
1889:
1876:
1863:
1862:
1846:circular sector
1842:
1840:Circular sector
1817:
1816:
1797:
1796:
1755:
1728:
1724:
1714:
1713:
1660:
1656:
1630:
1622:
1621:
1591:
1590:
1571:
1570:
1534:
1528:
1524:
1506:
1501:
1500:
1469:
1454:
1444:
1437:
1436:
1406:
1405:
1386:
1385:
1344:
1343:
1303:
1295:
1294:
1256:
1255:
1236:
1235:
1203:
1190:
1181:
1177:
1166:
1165:
1136:
1123:
1116:
1115:
1088:
1075:
1054:
1046:
1045:
1014:
1006:
1005:
986:
985:
966:
965:
928:
927:
924:lateral surface
920:
892:
891:
868:
867:
844:
831:
824:
823:
816:
784:
760:
759:
735:
727:
726:
715:
710:
698:
656:
624:
616:
615:
585:
564:
563:
542:
541:
540:and the height
520:
515:
514:
495:
494:
488:
483:
460:elliptical cone
427:Acta Eruditorum
368:
353:projective cone
315:at right angles
265:conical surface
261:lateral surface
257:two-dimensional
210:geometric shape
191:
158:
155:
138:
129:
127:
114:
75:
42:
39:Conical surface
35:
28:
27:Geometric shape
23:
22:
15:
12:
11:
5:
3910:
3908:
3900:
3899:
3894:
3884:
3883:
3880:
3879:
3873:
3868:
3863:
3856:
3837:
3818:
3797:
3796:External links
3794:
3793:
3792:
3781:Addison-Wesley
3770:
3767:
3764:
3763:
3747:
3730:
3728:, p. 583)
3718:
3711:
3691:
3684:
3672:(2013-11-11).
3661:
3654:
3631:
3606:
3588:
3581:
3552:
3551:
3549:
3546:
3545:
3544:
3539:
3534:
3529:
3524:
3519:
3517:List of shapes
3514:
3509:
3504:
3499:
3494:
3489:
3482:
3479:
3433:
3428:
3391:
3388:
3325:
3322:
3307:
3306:
3282:
3274:
3270:
3266:
3261:
3257:
3253:
3248:
3244:
3234:with equation
3224:
3223:
3212:
3207:
3203:
3199:
3192:
3188:
3182:
3178:
3172:
3165:
3161:
3155:
3151:
3113:
3110:
3093:
3090:
3087:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3032:
3031:
3020:
3017:
3014:
3010:
3006:
3002:
2997:
2993:
2989:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2951:
2938:
2934:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2883:
2879:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2828:
2825:
2822:
2819:
2816:
2813:
2789:
2786:
2766:
2755:
2754:
2742:
2737:
2733:
2729:
2726:
2723:
2720:
2715:
2711:
2707:
2702:
2698:
2694:
2691:
2688:
2685:
2682:
2677:
2673:
2669:
2664:
2660:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2615:
2614:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2513:
2510:
2507:
2504:
2501:
2486:
2483:
2480:
2477:
2474:
2471:
2451:
2448:
2445:
2442:
2439:
2419:
2416:
2413:
2410:
2407:
2396:
2395:
2383:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2327:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2276:
2256:
2253:
2244:and aperture
2233:
2209:
2205:
2202:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2150:
2149:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2063:
2060:
2059:
2058:
2057:
2056:
2040:
2036:
2032:
2027:
2023:
2017:
2014:
2011:
2005:
2000:
1997:
1992:
1989:
1976:
1975:
1970:central angle
1967:
1966:
1965:
1964:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1922:
1921:
1914:
1913:
1912:
1911:
1896:
1892:
1888:
1883:
1879:
1873:
1870:
1857:
1856:
1841:
1838:
1837:
1836:
1835:is the height.
1824:
1804:
1793:
1792:
1791:
1777:
1774:
1769:
1766:
1761:
1758:
1751:
1748:
1743:
1740:
1735:
1731:
1727:
1721:
1711:
1699:
1693:
1690:
1685:
1682:
1679:
1674:
1671:
1666:
1663:
1659:
1653:
1650:
1645:
1642:
1637:
1633:
1629:
1616:
1615:
1611:
1610:
1598:
1578:
1567:
1566:
1565:
1553:
1549:
1546:
1540:
1537:
1533:
1527:
1522:
1517:
1514:
1509:
1495:
1494:
1493:
1492:
1479:
1475:
1472:
1466:
1460:
1457:
1451:
1447:
1431:
1430:
1426:
1425:
1413:
1393:
1382:
1381:
1380:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1338:
1337:
1336:
1335:
1324:
1321:
1318:
1315:
1310:
1306:
1302:
1289:
1288:
1276:
1275:
1274:is the height.
1263:
1243:
1232:
1231:
1230:
1218:
1210:
1206:
1202:
1197:
1193:
1187:
1184:
1180:
1176:
1173:
1160:
1159:
1143:
1139:
1135:
1130:
1126:
1112:
1111:
1110:
1095:
1091:
1087:
1082:
1078:
1072:
1069:
1066:
1061:
1057:
1053:
1040:
1039:
1021:
1017:
1013:
993:
973:
953:
950:
947:
944:
941:
938:
935:
919:
916:
899:
875:
851:
847:
843:
838:
834:
815:
812:
811:
810:
799:
796:
791:
787:
783:
778:
775:
770:
767:
742:
738:
734:
714:
711:
709:
706:
702:center of mass
697:
696:Center of mass
694:
663:
659:
652:
649:
643:
640:
637:
631:
627:
623:
612:
611:
600:
597:
592:
588:
582:
579:
574:
571:
549:
527:
523:
502:
487:
484:
482:
479:
444:truncated cone
408:is called the
367:
364:
336:A cone with a
299:right circular
245:quadratic form
171:
170:
153:
147:
146:
125:
119:
118:
111:
109:Symmetry group
105:
104:
101:
95:
94:
91:
85:
84:
81:
77:
76:
70:and its angle
57:
49:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3909:
3898:
3895:
3893:
3890:
3889:
3887:
3877:
3874:
3872:
3869:
3867:
3864:
3861:
3860:Spinning Cone
3857:
3852:
3851:
3846:
3843:
3838:
3833:
3832:
3827:
3826:"Double Cone"
3824:
3819:
3814:
3813:
3808:
3805:
3800:
3799:
3795:
3790:
3786:
3782:
3778:
3773:
3772:
3768:
3760:
3756:
3755:G. B. Halsted
3751:
3748:
3743:
3742:
3734:
3731:
3727:
3722:
3719:
3714:
3712:9781931914598
3708:
3704:
3703:
3695:
3692:
3687:
3685:9780387226767
3681:
3677:
3676:
3671:
3665:
3662:
3657:
3655:9781285965901
3651:
3647:
3646:
3638:
3636:
3632:
3626:
3625:
3620:
3617:
3610:
3607:
3603:
3602:
3595:
3593:
3589:
3584:
3582:9780412990410
3578:
3574:
3573:
3568:
3562:
3560:
3558:
3554:
3547:
3543:
3540:
3538:
3537:Ruled surface
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3480:
3478:
3476:
3471:
3469:
3465:
3461:
3458:, the vector
3457:
3453:
3449:
3431:
3417:
3414:
3410:
3407:
3403:
3397:
3389:
3387:
3384:
3382:
3381:axial pencils
3378:
3377:Steiner conic
3374:
3373:G. B. Halsted
3371:According to
3369:
3367:
3363:
3359:
3355:
3351:
3347:
3339:
3335:
3330:
3323:
3321:
3319:
3314:
3312:
3304:
3303:plane section
3300:
3299:
3298:
3296:
3295:conic section
3280:
3272:
3268:
3264:
3259:
3255:
3251:
3246:
3242:
3233:
3229:
3210:
3205:
3201:
3197:
3190:
3186:
3180:
3176:
3170:
3163:
3159:
3153:
3149:
3139:
3138:
3137:
3135:
3131:
3130:elliptic cone
3127:
3118:
3112:Elliptic cone
3111:
3109:
3107:
3091:
3088:
3085:
3062:
3059:
3056:
3053:
3050:
3044:
3041:
3018:
3015:
3012:
3004:
2991:
2983:
2980:
2977:
2974:
2971:
2965:
2959:
2952:
2936:
2928:
2925:
2922:
2913:
2910:
2907:
2898:
2895:
2892:
2886:
2881:
2873:
2870:
2867:
2861:
2855:
2849:
2842:
2841:
2840:
2826:
2823:
2817:
2811:
2803:
2787:
2784:
2764:
2740:
2735:
2727:
2724:
2721:
2713:
2709:
2705:
2700:
2692:
2689:
2686:
2675:
2671:
2667:
2662:
2658:
2651:
2645:
2642:
2639:
2636:
2633:
2627:
2620:
2619:
2618:
2601:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2562:
2559:
2556:
2553:
2550:
2544:
2534:
2533:
2532:
2530:
2525:
2508:
2505:
2502:
2481:
2478:
2475:
2472:
2446:
2443:
2440:
2417:
2414:
2411:
2408:
2405:
2381:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2325:
2321:
2315:
2312:
2309:
2306:
2303:
2297:
2290:
2289:
2288:
2274:
2254:
2251:
2231:
2222:
2203:
2200:
2177:
2174:
2171:
2168:
2162:
2159:
2136:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2094:
2088:
2085:
2082:
2076:
2069:
2068:
2067:
2062:Equation form
2061:
2038:
2034:
2030:
2025:
2021:
2015:
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319:conic section
316:
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225:line segments
221:
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90:
86:
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69:
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62:, its height
61:
55:
50:
45:
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33:
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3609:
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3571:
3567:James, R. C.
3472:
3463:
3459:
3455:
3451:
3447:
3416:vector space
3408:
3399:
3385:
3370:
3343:
3315:
3308:
3302:
3231:
3228:affine image
3225:
3129:
3123:
3104:denotes the
3033:
2756:
2616:
2526:
2397:
2223:
2151:
2065:
1971:
1918:
1853:
1843:
921:
918:Surface area
817:
814:Slant height
722:
718:
716:
699:
689:
613:
489:
469:
468:
459:
458:
452:
443:
442:
440:
422:
410:
409:
405:
404:, the angle
397:
393:
388:
387:
381:
369:
357:
346:
335:
310:
302:
298:
292:
281:
276:
272:
269:
260:
253:solid object
222:
202:
200:
165:
162:
142:
133:
123:Surface area
83:Solid figure
71:
67:
63:
59:
3512:Hyperboloid
3402:convex cone
3358:right angle
3106:dot product
2430:range over
1917:arc length
721:and height
475:visual hull
349:convex cone
273:double cone
99:Euler char.
3886:Categories
3876:Cut a Cone
3769:References
3598:Grünbaum,
3502:Democritus
3406:convex set
1974:in radians
465:elliptical
449:truncation
411:half-angle
229:half-lines
3850:MathWorld
3831:MathWorld
3812:MathWorld
3761:, page 20
3624:MathWorld
3396:Hypercone
3232:unit cone
3226:It is an
3089:⋅
3019:θ
3016:
2984:−
2978:⋅
2929:θ
2926:
2911:⋅
2896:⋅
2887:−
2871:⋅
2804:equation
2788:θ
2728:θ
2725:
2706:−
2693:θ
2690:
2593:≤
2581:≥
2569:≤
2482:π
2447:θ
2369:
2360:
2345:
2336:
2255:θ
2204:∈
2178:π
2163:∈
2160:θ
2125:θ
2122:
2110:θ
2107:
2083:θ
2013:π
1988:φ
1949:π
1803:θ
1773:−
1765:θ
1760:
1747:θ
1742:
1726:π
1720:−
1689:θ
1684:
1670:θ
1665:
1649:θ
1644:
1628:π
1597:ℓ
1548:ℓ
1539:π
1474:ℓ
1459:π
1412:ℓ
1365:ℓ
1350:π
1323:ℓ
1317:π
1301:π
1172:π
1068:π
1052:π
1012:π
992:ℓ
952:ℓ
946:π
782:π
733:π
622:∫
467:base. A
447:; if the
373:directrix
338:polygonal
3897:Surfaces
3789:76087042
3481:See also
3350:cylinder
3338:cylinder
2529:implicit
866:, where
389:aperture
303:circular
301:, where
295:geometry
3757:(1906)
3527:Quadric
3411:in the
3132:is the
3124:In the
1852:radius
886:is the
454:frustum
342:pyramid
3807:"Cone"
3787:
3709:
3682:
3652:
3619:"Cone"
3579:
3487:Bicone
3462:is in
3354:arctan
3278:
3078:, and
3034:where
2839:where
2802:vector
2617:where
2398:where
2152:where
1795:where
1569:where
1384:where
1234:where
964:where
888:radius
820:circle
713:Volume
492:volume
486:Volume
457:. An
429:, 1734
402:optics
384:radius
307:circle
241:circle
218:vertex
151:Volume
3548:Notes
3134:locus
3128:, an
400:. In
351:or a
311:right
277:nappe
237:plane
233:lines
231:, or
205:is a
89:Faces
3785:LCCN
3707:ISBN
3680:ISBN
3650:ISBN
3577:ISBN
3413:real
3348:, a
3336:, a
3301:Any
1844:The
922:The
700:The
490:The
323:area
309:and
284:axis
282:The
214:apex
203:cone
115:O(2)
80:Type
47:Cone
3450:in
3344:In
3332:In
3313:).
3013:cos
2923:cos
2722:sin
2687:cos
2527:In
2366:sin
2357:tan
2342:cos
2333:tan
2119:sin
2104:cos
1757:sin
1739:sin
1681:sec
1662:tan
1641:tan
477:).
379:.)
216:or
168:)/3
3888::
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3828:.
3809:.
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3634:^
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3460:ax
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