407:
43:
318:
3318:
3106:
1268:
173:
669:– 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
182:
165:
359:
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
259:
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
1698:
1778:
2741:
1552:
2043:
2938:
3210:
2382:
2136:
1479:
1217:
3341:
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
1097:
2601:
663:
3018:
797:
423:
1612:
1898:
598:
314:, 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.
1145:
853:
3280:
2180:
1322:
2208:
3433:
1367:
951:
3375:"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'."
2449:
1951:
3065:
1021:
742:
3091:
2787:
2254:
1802:
2826:
2417:
1596:
1411:
991:
527:
2764:
2274:
2231:
1822:
1704:
1576:
1391:
1261:
1241:
971:
897:
873:
547:
500:
2511:
2484:
310:. 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
3354:
2612:
1491:
1972:
681:(can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.
693:
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.
2834:
3329:
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.
3131:
2282:
606:
248:
object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the
2061:
1427:
1156:
1036:
2526:
2944:
3699:
3672:
3642:
3569:
97:
3298:
Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see
275:
of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a
240:, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a
1693:{\displaystyle \pi h^{2}\tan {\frac {\theta }{2}}\left(\tan {\frac {\theta }{2}}+\sec {\frac {\theta }{2}}\right)}
3456:
993:
is the slant height of the cone. The surface area of the bottom circle of a cone is the same as for any circle,
750:
3455:. In this context, the analogues of circular cones are not usually special; in fact one is often interested in
603:
In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral
406:
1853:
554:
381:
of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle
462:
is the surface created by the set of lines passing through a vertex and every point on a boundary (also see
1106:
814:
201:
that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the
3226:
2144:
1285:
3114:
3880:
2185:
674:
3864:
3409:
306:
to its plane. If the cone is right circular the intersection of a plane with the lateral surface is a
361:
3317:
1334:
1023:. Thus, the total surface area of a right circular cone can be expressed as each of the following:
918:
666:
42:
665:
Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying
2422:
317:
195:
1918:
3026:
2454:
996:
717:
3070:
2769:
2236:
3480:
2746:
More generally, a right circular cone with vertex at the origin, axis parallel to the vector
1787:
437:
237:
20:
2796:
228:
that does not contain the apex. Depending on the author, the base may be restricted to be a
3369:
2390:
1581:
1396:
976:
670:
505:
348:
87:
1773:{\displaystyle -{\frac {\pi h^{2}\sin {\frac {\theta }{2}}}{\sin {\frac {\theta }{2}}-1}}}
8:
3530:
3485:
3338:
3334:
3326:
3322:
3122:
900:
811:
of its base to the apex via a line segment along the surface of the cone. It is given by
272:
3728:
2749:
2259:
2216:
1807:
1561:
1376:
1246:
1226:
956:
882:
858:
532:
485:
330:
3859:
3854:
3885:
3833:
3830:
3811:
3792:
3773:
3695:
3668:
3638:
3604:
3565:
3495:
2517:
276:
206:
3658:
3589:
3520:
3490:
3463:
3372:(not in perspective) rather than the projective ranges used for the Steiner conic:
3350:
3299:
2736:{\displaystyle F(x,y,z)=(x^{2}+y^{2})(\cos \theta )^{2}-z^{2}(\sin \theta )^{2}.\,}
365:
225:
102:
3689:
3662:
3632:
3559:
3555:
3510:
3306:
2790:
1834:
912:
415:
341:
311:
253:
245:
224:
connecting a common point, the apex, to all of the points on a base that is in a
221:
202:
198:
77:
27:
3814:
1547:{\displaystyle \left({\frac {c}{2}}\right)\left({\frac {c}{2\pi }}+\ell \right)}
807:
The slant height of a right circular cone is the distance from any point on the
3848:
3769:
3505:
1271:
Total surface area of a right circular cone, given radius 𝑟 and slant height ℓ
690:
241:
233:
217:
3874:
3743:
3525:
3365:
3361:
3283:
2276:
coordinate axis and whose apex is the origin, is described parametrically as
2038:{\displaystyle \varphi ={\frac {L}{R}}={\frac {2\pi r}{\sqrt {r^{2}+h^{2}}}}}
307:
303:
3404:
213:
111:
3795:
3607:
3105:
2933:{\displaystyle F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \theta )^{2}}
3500:
3401:
3390:
3346:
3094:
463:
337:
3867:
An interactive demonstration of the intersection of a cone with a plane
3394:
3286:
is a conic section of the same type (ellipse, parabola,...), one gets:
3216:
502:
of any conic solid is one third of the product of the area of the base
430:
A cone with a region including its apex cut off by a plane is called a
422:
3838:
3819:
3800:
3612:
3384:
1103:(the area of the base plus the area of the lateral surface; the term
375:
of its base; often this is simply called the radius of the cone. The
3205:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=z^{2}.}
1267:
3389:
The definition of a cone may be extended to higher dimensions; see
3305:
The intersection of an elliptic cone with a concentric sphere is a
2377:{\displaystyle F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)}
283:
264:. Either half of a double cone on one side of the apex is called a
321:
Air traffic control tower in the shape of a cone, Sharjah
Airport.
3515:
453:
442:
326:
3777:
3475:
3342:
1837:
is obtained by unfolding the surface of one nappe of the cone:
876:
808:
480:
390:
372:
295:
229:
139:
336:
Depending on the context, "cone" may also mean specifically a
2131:{\displaystyle f(\theta ,h)=(h\cos \theta ,h\sin \theta ,h),}
172:
3631:
Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01).
1474:{\displaystyle {\frac {c^{2}}{4\pi }}+{\frac {c\ell }{2}}}
973:
is the radius of the circle at the bottom of the cone and
302:
means that the axis passes through the centre of the base
181:
164:
3828:
3564:. Springer Science & Business Media. pp. 74–75.
1212:{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}
3435:
is a cone (with apex at the origin) if for every vector
146:
126:
117:
634:
3688:
Blank, Brian E.; Krantz, Steven George (2006-01-01).
3667:. Springer Science & Business Media. Chapter 27.
3412:
3229:
3134:
3073:
3029:
2947:
2837:
2799:
2772:
2752:
2615:
2529:
2487:
2457:
2425:
2393:
2285:
2262:
2239:
2219:
2188:
2147:
2064:
1975:
1921:
1856:
1810:
1790:
1707:
1615:
1584:
1564:
1494:
1430:
1399:
1379:
1337:
1288:
1249:
1229:
1159:
1109:
1039:
999:
979:
959:
921:
885:
861:
817:
753:
720:
609:
557:
535:
508:
488:
440:
plane is parallel to the cone's base, it is called a
3809:
3764:
Protter, Murray H.; Morrey, Charles B. Jr. (1970),
3694:. Springer Science & Business Media. Chapter 8.
3466:, which is defined in arbitrary topological spaces.
2520:
form, the same solid is defined by the inequalities
1092:{\displaystyle \pi r^{2}+\pi r{\sqrt {r^{2}+h^{2}}}}
3630:
2596:{\displaystyle \{F(x,y,z)\leq 0,z\geq 0,z\leq h\},}
658:{\displaystyle \int x^{2}\,dx={\tfrac {1}{3}}x^{3}}
3427:
3274:
3204:
3085:
3059:
3012:
2932:
2820:
2781:
2758:
2735:
2595:
2505:
2478:
2443:
2411:
2376:
2268:
2248:
2225:
2202:
2174:
2130:
2037:
1945:
1892:
1816:
1796:
1772:
1692:
1590:
1570:
1546:
1473:
1405:
1385:
1361:
1316:
1255:
1235:
1211:
1139:
1091:
1015:
985:
965:
945:
891:
867:
847:
791:
736:
677:– more precisely, not all polyhedral pyramids are
657:
592:
541:
521:
494:
403:of the cone, to distinguish it from the aperture.
168:A right circular cone and an oblique circular cone
47:A right circular cone with the radius of its base
3872:
3013:{\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }
252:; if the lateral surface is unbounded, it is a
2055:The surface of a cone can be parameterized as
3790:
3763:
3714:
3602:
469:
176:A double cone (not shown infinitely extended)
2587:
2530:
371:The "base radius" of a circular cone is the
3687:
792:{\displaystyle V={\frac {1}{3}}\pi r^{2}h.}
3657:
3583:
3581:
3554:
3282:From the fact, that the affine image of a
41:
3733:. McGraw-Hill book Company, Incorporated.
3415:
2732:
2196:
899:is the height. This can be proved by the
623:
3727:Dowling, Linnaeus Wayland (1917-01-01).
3634:Elementary Geometry for College Students
3596:
3316:
3104:
2213:A right solid circular cone with height
1266:
421:
405:
316:
179:
171:
163:
3860:Lateral surface area of an oblique cone
3766:College Calculus with Analytic Geometry
3726:
3578:
3294:of an elliptic cone is a conic section.
1893:{\displaystyle R={\sqrt {r^{2}+h^{2}}}}
593:{\displaystyle V={\frac {1}{3}}A_{B}h.}
3873:
3349:. This is useful in the definition of
3312:
744:and so the formula for volume becomes
696:
354:
3829:
3810:
3791:
3603:
3364:, a cone is generated similarly to a
1140:{\displaystyle {\sqrt {r^{2}+h^{2}}}}
848:{\displaystyle {\sqrt {r^{2}+h^{2}}}}
673:. This is essentially the content of
426:A cone truncated by an inclined plane
3626:
3624:
3550:
3548:
3546:
3462:An even more general concept is the
3275:{\displaystyle x^{2}+y^{2}=z^{2}\ .}
2182:is the angle "around" the cone, and
2175:{\displaystyle \theta \in [0,2\pi )}
1317:{\displaystyle \pi r^{2}+\pi r\ell }
82:1 circular face and 1 conic surface
13:
3443:and every nonnegative real number
3378:
3109:An elliptical cone quadric surface
1828:
14:
3897:
3784:
3621:
3543:
2203:{\displaystyle h\in \mathbb {R} }
915:area of a right circular cone is
684:
347:Cones can also be generalized to
3428:{\displaystyle \mathbb {R} ^{n}}
3393:. In this case, one says that a
3353:, which require considering the
3100:
2210:is the "height" along the cone.
2050:
706:For a circular cone with radius
906:
802:
714:, the base is a circle of area
180:
3737:
3720:
3708:
3681:
3651:
3054:
3036:
2997:
2989:
2984:
2976:
2957:
2951:
2921:
2908:
2905:
2893:
2890:
2878:
2866:
2853:
2847:
2841:
2809:
2803:
2720:
2707:
2685:
2672:
2669:
2643:
2637:
2619:
2554:
2536:
2500:
2488:
2473:
2458:
2438:
2426:
2307:
2289:
2169:
2154:
2122:
2086:
2080:
2068:
1418:Circumference and slant height
1362:{\displaystyle \pi r(r+\ell )}
1356:
1344:
946:{\displaystyle LSA=\pi r\ell }
385:to the axis, the aperture is 2
282:In common usage in elementary
1:
3757:
3748:Synthetic Projective Geometry
3558:; James, Glenn (1992-07-31).
3368:only with a projectivity and
212:A cone is formed by a set of
7:
3664:Geometry: Euclid and Beyond
3469:
3125:of an equation of the form
3115:Cartesian coordinate system
2789:, is given by the implicit
2444:{\displaystyle [0,\theta )}
701:
10:
3902:
3715:Protter & Morrey (1970
3561:The Mathematics Dictionary
3382:
1946:{\displaystyle L=c=2\pi r}
470:Measurements and equations
286:, cones are assumed to be
25:
18:
3768:(2nd ed.), Reading:
3691:Calculus: Single Variable
3345:, in the limit forming a
3060:{\displaystyle u=(x,y,z)}
2479:{\displaystyle [0,2\pi )}
1578:is the circumference and
1016:{\displaystyle \pi r^{2}}
737:{\displaystyle \pi r^{2}}
474:
412:Problemata mathematica...
294:means that the base is a
236:in the plane, any closed
138:
110:
96:
86:
76:
68:
40:
35:
3593:, second edition, p. 23.
3536:
3086:{\displaystyle u\cdot d}
2782:{\displaystyle 2\theta }
2249:{\displaystyle 2\theta }
364:of a conic section, see
26:Not to be confused with
1797:{\displaystyle \theta }
1276:Radius and slant height
675:Hilbert's third problem
3429:
3330:
3276:
3219:of the right-circular
3206:
3110:
3087:
3061:
3014:
2934:
2822:
2821:{\displaystyle F(u)=0}
2783:
2760:
2737:
2597:
2507:
2480:
2445:
2413:
2378:
2270:
2250:
2227:
2204:
2176:
2132:
2039:
1947:
1894:
1818:
1804:is the apex angle and
1798:
1774:
1694:
1592:
1572:
1548:
1475:
1407:
1387:
1363:
1318:
1272:
1257:
1237:
1213:
1141:
1093:
1017:
987:
967:
947:
893:
869:
849:
793:
738:
659:
594:
543:
523:
496:
427:
419:
322:
238:one-dimensional figure
232:, any one-dimensional
187:
177:
169:
3481:Cone (linear algebra)
3430:
3383:Further information:
3320:
3277:
3207:
3108:
3088:
3062:
3015:
2935:
2823:
2784:
2761:
2738:
2598:
2508:
2481:
2446:
2414:
2412:{\displaystyle s,t,u}
2379:
2271:
2251:
2228:
2205:
2177:
2133:
2040:
1948:
1895:
1819:
1799:
1775:
1695:
1603:Apex angle and height
1593:
1591:{\displaystyle \ell }
1573:
1549:
1476:
1408:
1406:{\displaystyle \ell }
1388:
1364:
1319:
1270:
1258:
1238:
1214:
1142:
1094:
1018:
988:
986:{\displaystyle \ell }
968:
948:
894:
870:
850:
794:
739:
667:Cavalieri's principle
660:
595:
544:
524:
522:{\displaystyle A_{B}}
497:
425:
409:
320:
185:
175:
167:
21:Cone (disambiguation)
3637:. Cengage Learning.
3410:
3227:
3132:
3071:
3027:
2945:
2835:
2797:
2770:
2750:
2613:
2527:
2485:
2455:
2423:
2391:
2283:
2260:
2256:, whose axis is the
2237:
2217:
2186:
2145:
2062:
1973:
1919:
1854:
1808:
1788:
1705:
1613:
1598:is the slant height.
1582:
1562:
1492:
1428:
1413:is the slant height.
1397:
1377:
1335:
1286:
1247:
1227:
1157:
1147:is the slant height)
1107:
1037:
997:
977:
957:
919:
883:
859:
815:
751:
718:
671:method of exhaustion
607:
555:
533:
506:
486:
355:Further terminology
244:; otherwise it is a
19:For other uses, see
3730:Projective Geometry
3531:Translation of axes
3486:Cylinder (geometry)
3335:projective geometry
3323:projective geometry
3313:Projective geometry
901:Pythagorean theorem
697:Right circular cone
55:, its slant height
3834:"Generalized Cone"
3831:Weisstein, Eric W.
3812:Weisstein, Eric W.
3793:Weisstein, Eric W.
3605:Weisstein, Eric W.
3425:
3355:cylindrical conics
3331:
3272:
3202:
3111:
3083:
3057:
3010:
2930:
2818:
2779:
2756:
2733:
2593:
2503:
2476:
2441:
2409:
2374:
2266:
2246:
2223:
2200:
2172:
2128:
2035:
1943:
1890:
1814:
1794:
1770:
1690:
1588:
1568:
1544:
1471:
1403:
1393:is the radius and
1383:
1359:
1314:
1273:
1253:
1243:is the radius and
1233:
1209:
1137:
1089:
1013:
983:
963:
943:
889:
865:
845:
789:
734:
679:scissors congruent
655:
643:
590:
539:
519:
492:
452:is a cone with an
428:
420:
410:Illustration from
323:
188:
186:3D model of a cone
178:
170:
3881:Elementary shapes
3851:from Maths Is Fun
3659:Hartshorne, Robin
3496:Generalized conic
3351:degenerate conics
3268:
3184:
3157:
2759:{\displaystyle d}
2269:{\displaystyle z}
2226:{\displaystyle h}
2033:
2032:
1990:
1888:
1817:{\displaystyle h}
1768:
1759:
1741:
1683:
1664:
1643:
1571:{\displaystyle c}
1531:
1507:
1469:
1451:
1386:{\displaystyle r}
1256:{\displaystyle h}
1236:{\displaystyle r}
1202:
1135:
1087:
1027:Radius and height
966:{\displaystyle r}
892:{\displaystyle h}
868:{\displaystyle r}
843:
768:
642:
572:
542:{\displaystyle h}
495:{\displaystyle V}
349:higher dimensions
329:base is called a
277:circular symmetry
196:three-dimensional
162:
161:
3893:
3855:Paper model cone
3844:
3843:
3825:
3824:
3806:
3805:
3780:
3751:
3741:
3735:
3734:
3724:
3718:
3712:
3706:
3705:
3685:
3679:
3678:
3655:
3649:
3648:
3628:
3619:
3618:
3617:
3600:
3594:
3590:Convex Polytopes
3585:
3576:
3575:
3552:
3521:Rotation of axes
3464:topological cone
3457:polyhedral cones
3434:
3432:
3431:
3426:
3424:
3423:
3418:
3300:circular section
3281:
3279:
3278:
3273:
3266:
3265:
3264:
3252:
3251:
3239:
3238:
3211:
3209:
3208:
3203:
3198:
3197:
3185:
3183:
3182:
3173:
3172:
3163:
3158:
3156:
3155:
3146:
3145:
3136:
3092:
3090:
3089:
3084:
3066:
3064:
3063:
3058:
3019:
3017:
3016:
3011:
3000:
2992:
2987:
2979:
2939:
2937:
2936:
2931:
2929:
2928:
2874:
2873:
2827:
2825:
2824:
2819:
2788:
2786:
2785:
2780:
2765:
2763:
2762:
2757:
2742:
2740:
2739:
2734:
2728:
2727:
2706:
2705:
2693:
2692:
2668:
2667:
2655:
2654:
2602:
2600:
2599:
2594:
2513:, respectively.
2512:
2510:
2509:
2506:{\displaystyle }
2504:
2483:
2482:
2477:
2450:
2448:
2447:
2442:
2418:
2416:
2415:
2410:
2383:
2381:
2380:
2375:
2373:
2369:
2275:
2273:
2272:
2267:
2255:
2253:
2252:
2247:
2232:
2230:
2229:
2224:
2209:
2207:
2206:
2201:
2199:
2181:
2179:
2178:
2173:
2137:
2135:
2134:
2129:
2044:
2042:
2041:
2036:
2034:
2031:
2030:
2018:
2017:
2008:
2007:
1996:
1991:
1983:
1952:
1950:
1949:
1944:
1899:
1897:
1896:
1891:
1889:
1887:
1886:
1874:
1873:
1864:
1823:
1821:
1820:
1815:
1803:
1801:
1800:
1795:
1779:
1777:
1776:
1771:
1769:
1767:
1760:
1752:
1743:
1742:
1734:
1726:
1725:
1712:
1699:
1697:
1696:
1691:
1689:
1685:
1684:
1676:
1665:
1657:
1644:
1636:
1628:
1627:
1597:
1595:
1594:
1589:
1577:
1575:
1574:
1569:
1553:
1551:
1550:
1545:
1543:
1539:
1532:
1530:
1519:
1512:
1508:
1500:
1480:
1478:
1477:
1472:
1470:
1465:
1457:
1452:
1450:
1442:
1441:
1432:
1412:
1410:
1409:
1404:
1392:
1390:
1389:
1384:
1368:
1366:
1365:
1360:
1323:
1321:
1320:
1315:
1301:
1300:
1262:
1260:
1259:
1254:
1242:
1240:
1239:
1234:
1218:
1216:
1215:
1210:
1208:
1204:
1203:
1201:
1200:
1188:
1187:
1178:
1146:
1144:
1143:
1138:
1136:
1134:
1133:
1121:
1120:
1111:
1098:
1096:
1095:
1090:
1088:
1086:
1085:
1073:
1072:
1063:
1052:
1051:
1022:
1020:
1019:
1014:
1012:
1011:
992:
990:
989:
984:
972:
970:
969:
964:
952:
950:
949:
944:
898:
896:
895:
890:
879:of the base and
874:
872:
871:
866:
854:
852:
851:
846:
844:
842:
841:
829:
828:
819:
798:
796:
795:
790:
782:
781:
769:
761:
743:
741:
740:
735:
733:
732:
664:
662:
661:
656:
654:
653:
644:
635:
622:
621:
599:
597:
596:
591:
583:
582:
573:
565:
548:
546:
545:
540:
528:
526:
525:
520:
518:
517:
501:
499:
498:
493:
459:generalized cone
366:Dandelin spheres
184:
158:
149:
134:
129:
120:
105:
45:
33:
32:
3901:
3900:
3896:
3895:
3894:
3892:
3891:
3890:
3871:
3870:
3847:An interactive
3787:
3760:
3755:
3754:
3742:
3738:
3725:
3721:
3713:
3709:
3702:
3686:
3682:
3675:
3656:
3652:
3645:
3629:
3622:
3601:
3597:
3586:
3579:
3572:
3553:
3544:
3539:
3511:Pyrometric cone
3472:
3419:
3414:
3413:
3411:
3408:
3407:
3387:
3381:
3379:Generalizations
3315:
3307:spherical conic
3260:
3256:
3247:
3243:
3234:
3230:
3228:
3225:
3224:
3193:
3189:
3178:
3174:
3168:
3164:
3162:
3151:
3147:
3141:
3137:
3135:
3133:
3130:
3129:
3103:
3072:
3069:
3068:
3028:
3025:
3024:
2996:
2988:
2983:
2975:
2946:
2943:
2942:
2924:
2920:
2869:
2865:
2836:
2833:
2832:
2798:
2795:
2794:
2771:
2768:
2767:
2766:, and aperture
2751:
2748:
2747:
2723:
2719:
2701:
2697:
2688:
2684:
2663:
2659:
2650:
2646:
2614:
2611:
2610:
2528:
2525:
2524:
2486:
2456:
2453:
2452:
2424:
2421:
2420:
2392:
2389:
2388:
2317:
2313:
2284:
2281:
2280:
2261:
2258:
2257:
2238:
2235:
2234:
2218:
2215:
2214:
2195:
2187:
2184:
2183:
2146:
2143:
2142:
2063:
2060:
2059:
2053:
2026:
2022:
2013:
2009:
1997:
1995:
1982:
1974:
1971:
1970:
1920:
1917:
1916:
1882:
1878:
1869:
1865:
1863:
1855:
1852:
1851:
1835:circular sector
1831:
1829:Circular sector
1809:
1806:
1805:
1789:
1786:
1785:
1751:
1744:
1733:
1721:
1717:
1713:
1711:
1706:
1703:
1702:
1675:
1656:
1649:
1645:
1635:
1623:
1619:
1614:
1611:
1610:
1583:
1580:
1579:
1563:
1560:
1559:
1523:
1518:
1517:
1513:
1499:
1495:
1493:
1490:
1489:
1458:
1456:
1443:
1437:
1433:
1431:
1429:
1426:
1425:
1398:
1395:
1394:
1378:
1375:
1374:
1336:
1333:
1332:
1296:
1292:
1287:
1284:
1283:
1248:
1245:
1244:
1228:
1225:
1224:
1196:
1192:
1183:
1179:
1177:
1170:
1166:
1158:
1155:
1154:
1129:
1125:
1116:
1112:
1110:
1108:
1105:
1104:
1081:
1077:
1068:
1064:
1062:
1047:
1043:
1038:
1035:
1034:
1007:
1003:
998:
995:
994:
978:
975:
974:
958:
955:
954:
920:
917:
916:
913:lateral surface
909:
884:
881:
880:
860:
857:
856:
837:
833:
824:
820:
818:
816:
813:
812:
805:
777:
773:
760:
752:
749:
748:
728:
724:
719:
716:
715:
704:
699:
687:
649:
645:
633:
617:
613:
608:
605:
604:
578:
574:
564:
556:
553:
552:
534:
531:
530:
529:and the height
513:
509:
507:
504:
503:
487:
484:
483:
477:
472:
449:elliptical cone
416:Acta Eruditorum
357:
342:projective cone
304:at right angles
254:conical surface
250:lateral surface
246:two-dimensional
199:geometric shape
147:
144:
127:
118:
116:
103:
64:
31:
28:Conical surface
24:
17:
16:Geometric shape
12:
11:
5:
3899:
3889:
3888:
3883:
3869:
3868:
3862:
3857:
3852:
3845:
3826:
3807:
3786:
3785:External links
3783:
3782:
3781:
3770:Addison-Wesley
3759:
3756:
3753:
3752:
3736:
3719:
3717:, p. 583)
3707:
3700:
3680:
3673:
3661:(2013-11-11).
3650:
3643:
3620:
3595:
3577:
3570:
3541:
3540:
3538:
3535:
3534:
3533:
3528:
3523:
3518:
3513:
3508:
3506:List of shapes
3503:
3498:
3493:
3488:
3483:
3478:
3471:
3468:
3422:
3417:
3380:
3377:
3314:
3311:
3296:
3295:
3271:
3263:
3259:
3255:
3250:
3246:
3242:
3237:
3233:
3223:with equation
3213:
3212:
3201:
3196:
3192:
3188:
3181:
3177:
3171:
3167:
3161:
3154:
3150:
3144:
3140:
3102:
3099:
3082:
3079:
3076:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3021:
3020:
3009:
3006:
3003:
2999:
2995:
2991:
2986:
2982:
2978:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2940:
2927:
2923:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2872:
2868:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2817:
2814:
2811:
2808:
2805:
2802:
2778:
2775:
2755:
2744:
2743:
2731:
2726:
2722:
2718:
2715:
2712:
2709:
2704:
2700:
2696:
2691:
2687:
2683:
2680:
2677:
2674:
2671:
2666:
2662:
2658:
2653:
2649:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2604:
2603:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2502:
2499:
2496:
2493:
2490:
2475:
2472:
2469:
2466:
2463:
2460:
2440:
2437:
2434:
2431:
2428:
2408:
2405:
2402:
2399:
2396:
2385:
2384:
2372:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2316:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2265:
2245:
2242:
2233:and aperture
2222:
2198:
2194:
2191:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2139:
2138:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2052:
2049:
2048:
2047:
2046:
2045:
2029:
2025:
2021:
2016:
2012:
2006:
2003:
2000:
1994:
1989:
1986:
1981:
1978:
1965:
1964:
1959:central angle
1956:
1955:
1954:
1953:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1911:
1910:
1903:
1902:
1901:
1900:
1885:
1881:
1877:
1872:
1868:
1862:
1859:
1846:
1845:
1830:
1827:
1826:
1825:
1824:is the height.
1813:
1793:
1782:
1781:
1780:
1766:
1763:
1758:
1755:
1750:
1747:
1740:
1737:
1732:
1729:
1724:
1720:
1716:
1710:
1700:
1688:
1682:
1679:
1674:
1671:
1668:
1663:
1660:
1655:
1652:
1648:
1642:
1639:
1634:
1631:
1626:
1622:
1618:
1605:
1604:
1600:
1599:
1587:
1567:
1556:
1555:
1554:
1542:
1538:
1535:
1529:
1526:
1522:
1516:
1511:
1506:
1503:
1498:
1484:
1483:
1482:
1481:
1468:
1464:
1461:
1455:
1449:
1446:
1440:
1436:
1420:
1419:
1415:
1414:
1402:
1382:
1371:
1370:
1369:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1327:
1326:
1325:
1324:
1313:
1310:
1307:
1304:
1299:
1295:
1291:
1278:
1277:
1265:
1264:
1263:is the height.
1252:
1232:
1221:
1220:
1219:
1207:
1199:
1195:
1191:
1186:
1182:
1176:
1173:
1169:
1165:
1162:
1149:
1148:
1132:
1128:
1124:
1119:
1115:
1101:
1100:
1099:
1084:
1080:
1076:
1071:
1067:
1061:
1058:
1055:
1050:
1046:
1042:
1029:
1028:
1010:
1006:
1002:
982:
962:
942:
939:
936:
933:
930:
927:
924:
908:
905:
888:
864:
840:
836:
832:
827:
823:
804:
801:
800:
799:
788:
785:
780:
776:
772:
767:
764:
759:
756:
731:
727:
723:
703:
700:
698:
695:
691:center of mass
686:
685:Center of mass
683:
652:
648:
641:
638:
632:
629:
626:
620:
616:
612:
601:
600:
589:
586:
581:
577:
571:
568:
563:
560:
538:
516:
512:
491:
476:
473:
471:
468:
433:truncated cone
397:is called the
356:
353:
325:A cone with a
288:right circular
234:quadratic form
160:
159:
142:
136:
135:
114:
108:
107:
100:
98:Symmetry group
94:
93:
90:
84:
83:
80:
74:
73:
70:
66:
65:
59:and its angle
46:
38:
37:
15:
9:
6:
4:
3:
2:
3898:
3887:
3884:
3882:
3879:
3878:
3876:
3866:
3863:
3861:
3858:
3856:
3853:
3850:
3849:Spinning Cone
3846:
3841:
3840:
3835:
3832:
3827:
3822:
3821:
3816:
3815:"Double Cone"
3813:
3808:
3803:
3802:
3797:
3794:
3789:
3788:
3779:
3775:
3771:
3767:
3762:
3761:
3749:
3745:
3744:G. B. Halsted
3740:
3732:
3731:
3723:
3716:
3711:
3703:
3701:9781931914598
3697:
3693:
3692:
3684:
3676:
3674:9780387226767
3670:
3666:
3665:
3660:
3654:
3646:
3644:9781285965901
3640:
3636:
3635:
3627:
3625:
3615:
3614:
3609:
3606:
3599:
3592:
3591:
3584:
3582:
3573:
3571:9780412990410
3567:
3563:
3562:
3557:
3551:
3549:
3547:
3542:
3532:
3529:
3527:
3526:Ruled surface
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3473:
3467:
3465:
3460:
3458:
3454:
3450:
3447:, the vector
3446:
3442:
3438:
3420:
3406:
3403:
3399:
3396:
3392:
3386:
3376:
3373:
3371:
3370:axial pencils
3367:
3366:Steiner conic
3363:
3362:G. B. Halsted
3360:According to
3358:
3356:
3352:
3348:
3344:
3340:
3336:
3328:
3324:
3319:
3310:
3308:
3303:
3301:
3293:
3292:plane section
3289:
3288:
3287:
3285:
3284:conic section
3269:
3261:
3257:
3253:
3248:
3244:
3240:
3235:
3231:
3222:
3218:
3199:
3194:
3190:
3186:
3179:
3175:
3169:
3165:
3159:
3152:
3148:
3142:
3138:
3128:
3127:
3126:
3124:
3120:
3119:elliptic cone
3116:
3107:
3101:Elliptic cone
3098:
3096:
3080:
3077:
3074:
3051:
3048:
3045:
3042:
3039:
3033:
3030:
3007:
3004:
3001:
2993:
2980:
2972:
2969:
2966:
2963:
2960:
2954:
2948:
2941:
2925:
2917:
2914:
2911:
2902:
2899:
2896:
2887:
2884:
2881:
2875:
2870:
2862:
2859:
2856:
2850:
2844:
2838:
2831:
2830:
2829:
2815:
2812:
2806:
2800:
2792:
2776:
2773:
2753:
2729:
2724:
2716:
2713:
2710:
2702:
2698:
2694:
2689:
2681:
2678:
2675:
2664:
2660:
2656:
2651:
2647:
2640:
2634:
2631:
2628:
2625:
2622:
2616:
2609:
2608:
2607:
2590:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2551:
2548:
2545:
2542:
2539:
2533:
2523:
2522:
2521:
2519:
2514:
2497:
2494:
2491:
2470:
2467:
2464:
2461:
2435:
2432:
2429:
2406:
2403:
2400:
2397:
2394:
2370:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2314:
2310:
2304:
2301:
2298:
2295:
2292:
2286:
2279:
2278:
2277:
2263:
2243:
2240:
2220:
2211:
2192:
2189:
2166:
2163:
2160:
2157:
2151:
2148:
2125:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2083:
2077:
2074:
2071:
2065:
2058:
2057:
2056:
2051:Equation form
2027:
2023:
2019:
2014:
2010:
2004:
2001:
1998:
1992:
1987:
1984:
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489:
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413:
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308:conic section
305:
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293:
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285:
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214:line segments
210:
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156:
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137:
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95:
91:
89:
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79:
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67:
62:
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51:, its height
50:
44:
39:
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3556:James, R. C.
3461:
3452:
3448:
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3436:
3405:vector space
3397:
3388:
3374:
3359:
3332:
3304:
3297:
3291:
3220:
3217:affine image
3214:
3118:
3112:
3093:denotes the
3022:
2745:
2605:
2515:
2386:
2212:
2140:
2054:
1960:
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1842:
1832:
910:
907:Surface area
806:
803:Slant height
711:
707:
705:
688:
678:
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478:
458:
457:
448:
447:
441:
432:
431:
429:
411:
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398:
394:
393:, the angle
386:
382:
377:
376:
370:
358:
346:
335:
324:
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291:
287:
281:
270:
265:
261:
258:
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242:solid object
211:
191:
189:
154:
151:
131:
122:
112:Surface area
72:Solid figure
60:
56:
52:
48:
3501:Hyperboloid
3391:convex cone
3347:right angle
3095:dot product
2419:range over
1906:arc length
710:and height
464:visual hull
338:convex cone
262:double cone
88:Euler char.
3875:Categories
3865:Cut a Cone
3758:References
3587:Grünbaum,
3491:Democritus
3395:convex set
1963:in radians
454:elliptical
438:truncation
400:half-angle
218:half-lines
3839:MathWorld
3820:MathWorld
3801:MathWorld
3750:, page 20
3613:MathWorld
3385:Hypercone
3221:unit cone
3215:It is an
3078:⋅
3008:θ
3005:
2973:−
2967:⋅
2918:θ
2915:
2900:⋅
2885:⋅
2876:−
2860:⋅
2793:equation
2777:θ
2717:θ
2714:
2695:−
2682:θ
2679:
2582:≤
2570:≥
2558:≤
2471:π
2436:θ
2358:
2349:
2334:
2325:
2244:θ
2193:∈
2167:π
2152:∈
2149:θ
2114:θ
2111:
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2072:θ
2002:π
1977:φ
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1678:θ
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1586:ℓ
1537:ℓ
1528:π
1463:ℓ
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1401:ℓ
1354:ℓ
1339:π
1312:ℓ
1306:π
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1057:π
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1001:π
981:ℓ
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935:π
771:π
722:π
611:∫
456:base. A
436:; if the
362:directrix
327:polygonal
3886:Surfaces
3778:76087042
3470:See also
3339:cylinder
3327:cylinder
2518:implicit
855:, where
378:aperture
292:circular
290:, where
284:geometry
3746:(1906)
3516:Quadric
3400:in the
3121:is the
3113:In the
1841:radius
875:is the
443:frustum
331:pyramid
3796:"Cone"
3776:
3698:
3671:
3641:
3608:"Cone"
3568:
3476:Bicone
3451:is in
3343:arctan
3267:
3067:, and
3023:where
2828:where
2791:vector
2606:where
2387:where
2141:where
1784:where
1558:where
1373:where
1223:where
953:where
877:radius
809:circle
702:Volume
481:volume
475:Volume
446:. An
418:, 1734
391:optics
373:radius
296:circle
230:circle
207:vertex
140:Volume
3537:Notes
3123:locus
3117:, an
389:. In
340:or a
300:right
266:nappe
226:plane
222:lines
220:, or
194:is a
78:Faces
3774:LCCN
3696:ISBN
3669:ISBN
3639:ISBN
3566:ISBN
3402:real
3337:, a
3325:, a
3290:Any
1833:The
911:The
689:The
479:The
312:area
298:and
273:axis
271:The
203:apex
192:cone
104:O(2)
69:Type
36:Cone
3439:in
3333:In
3321:In
3302:).
3002:cos
2912:cos
2711:sin
2676:cos
2516:In
2355:sin
2346:tan
2331:cos
2322:tan
2108:sin
2093:cos
1746:sin
1728:sin
1670:sec
1651:tan
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466:).
368:.)
205:or
157:)/3
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