2728:
2780:
3084:
3070:
3056:
3100:
3091:
3077:
3049:
1685:
3063:
699:
508:. If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.
375:
1880:
454:
3025:
1394:
496:
3032:
3114:
2983:
2976:
2969:
3018:
3004:
2990:
278:
47:
3011:
2997:
2816:
treatment) on circular cylinders is that a circular base is the only type of geometric figure for which this technique works with the use of only elementary considerations (no appeal to calculus or more advanced mathematics). Terminology about prisms and cylinders is identical. Thus, for example, since a
2764:. If the cone is a quadratic cone, the plane at infinity (which passes through the vertex) can intersect the cone at two real lines, a single real line (actually a coincident pair of lines), or only at the vertex. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively.
2815:
and cylinders simultaneously. Formulas for surface area and volume are derived from the corresponding formulas for prisms by using inscribed and circumscribed prisms and then letting the number of sides of the prism increase without bound. One reason for the early emphasis (and sometimes exclusive
503:
For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the
1062:
1835:. A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line. Such cylinders have, at times, been referred to as
621:
2151:
917:
1557:
2494:
2642:
2363:
928:
1672:
2233:
1301:
1732:
that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius
933:
549:
2016:
365:. A cylinder of revolution is a right circular cylinder. The height of a cylinder of revolution is the length of the generating line segment. The line that the segment is revolved about is called the
408:
often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an
274:, not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of the generatrix is an element of the cylindrical surface.
544:
3975:
2045:
1841:. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder. Thus, this definition may be rephrased to say that a cylinder is any
429:
generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.
782:
723:
In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having
3559:
1444:
1887:
When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by
2723:
1380:
684:
2562:
2417:
2571:
2292:
1179:
2520:
2408:
2275:
2034:
being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriate
1581:
3968:
1390:
is the perimeter of a right section of the cylinder. This produces the previous formula for lateral area when the cylinder is a right circular cylinder.
2156:
1186:
1183:
The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore
488:. If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a
2873:
1890:
303:. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a
3961:
3875:
3773:
1057:{\displaystyle {\begin{aligned}V&=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{r}s\,\,ds\,d\phi \,dz\\&=\pi \,r^{2}\,h.\end{aligned}}}
3941:
761:). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive
143:
3556:
4258:
3947:
3897:
3824:
3688:
3589:
2962:
482:
A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a
92:
2820:
is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called a
3517:
3497:
3485:
3465:
3456:
3436:
3427:
3407:
3398:
3378:
3369:
3349:
3340:
3320:
3311:
3291:
3282:
3262:
3253:
3233:
3224:
3204:
3195:
3175:
3502:
3507:
3475:
3446:
3417:
3388:
3359:
3330:
3301:
3272:
3243:
3214:
3185:
3412:
3470:
3441:
724:
3512:
3480:
3451:
3422:
3393:
3383:
3364:
3354:
3335:
3325:
3306:
3296:
3277:
3267:
3248:
3238:
3219:
3209:
3190:
3180:
211:. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the
492:(parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.
4114:
2866:
299:. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an
4253:
2659:
1709:
obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a
1693:
512:
2683:
1350:
650:
3839:
313:
figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a
31:
4150:
2739:
is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
2727:
1149:
466:
1416:) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel
692:
4223:
4154:
2783:
1717:
922:
399:
213:
203:
surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to
1674:
Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.
3953:
2859:
136:
4090:
4032:
2286:
1417:
310:
247:
188:
1340:, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).
4212:
4187:
3121:
616:{\displaystyle {\begin{aligned}e&=\cos \alpha ,\\a&={\frac {r}{\sin \alpha }}.\end{aligned}}}
330:
82:
2779:
4182:
4176:
2749:
2732:
2254:
426:
230:
169:
1330:. Equivalently, for a given surface area, the right circular cylinder with the largest volume has
258:
in a plane not parallel to the given line. Any line in this family of parallel lines is called an
4077:
2937:
2846:
2541:
326:
196:
1688:
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases
4263:
4248:
4056:
3924:
3893:
3871:
3820:
3769:
3684:
3678:
3585:
2950:
2942:
2761:
2384:
2146:{\displaystyle A\left(x+{\frac {D}{2A}}\right)^{2}+B\left(y+{\frac {E}{2B}}\right)^{2}=\rho ,}
2042:
does not appear and the general equation of this type of degenerate quadric can be written as
1146:
does not include either top or bottom elements, and therefore has surface area (lateral area)
1085:
of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:
75:
2499:
4038:
3083:
3069:
3055:
2927:
2917:
2907:
2895:
2853:
2812:
2768:
2035:
462:
173:
97:
3099:
3090:
3076:
2393:
2365:
This equation of an elliptic cylinder is a generalization of the equation of the ordinary,
2260:
4140:
3563:
3537:
3167:
2932:
2922:
2912:
2828:
2795:
2757:
2753:
2736:
1873:
912:{\displaystyle V=\int _{0}^{h}A(x)dx=\int _{0}^{h}\pi abdx=\pi ab\int _{0}^{h}dx=\pi abh.}
521:
511:
In the case of a right circular cylinder with a cylindric section that is an ellipse, the
290:
4102:
1552:{\displaystyle V=\pi \left(R^{2}-r^{2}\right)h=2\pi \left({\frac {R+r}{2}}\right)h(R-r).}
1347:, of a circular cylinder, which need not be a right cylinder, is more generally given by
3048:
3532:
3062:
1819:. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
1684:
286:
251:
157:
71:
698:
4242:
4203:
4159:
4145:
4043:
3886:
3843:
3643:
1842:
489:
472:
349:
374:
4050:
2489:{\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=-1,}
1714:
1320:
For a given volume, the right circular cylinder with the smallest surface area has
1082:
415:
356:
184:
106:
3024:
2637:{\displaystyle \left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1.}
2358:{\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.}
1879:
471:. The cylindric section by a plane that contains two elements of a cylinder is a
3031:
2019:
495:
453:
255:
161:
2982:
2975:
2968:
3927:
3017:
3003:
2989:
1706:
1393:
339:. In some elementary treatments, a cylinder always means a circular cylinder.
263:
4218:
4107:
3932:
3674:
3648:
3113:
2836:
1857:
476:
2961:
925:, the volume of a right circular cylinder can be calculated by integration
17:
2888:
2808:
1853:
1721:
1314:
270:, a cylindrical surface is that surface traced out by a line, called the
225:
The definitions and results in this section are taken from the 1913 text
192:
1849:
1667:{\displaystyle A=2\pi \left(R+r\right)h+2\pi \left(R^{2}-r^{2}\right).}
505:
461:
A cylindric section is the intersection of a cylinder's surface with a
1420:
bases perpendicular to the cylinders' common axis, as in the diagram.
436:
has a height much greater than its diameter, whereas a short and wide
277:
4003:
2832:
1710:
644:
631:
419:
200:
195:. The shift in the basic meaning—solid versus surface (as in a solid
177:
149:
118:
1117:
The area of the top and bottom bases is the same, and is called the
688:
This formula holds whether or not the cylinder is a right cylinder.
371:
of the cylinder and it passes through the centers of the two bases.
46:
3010:
2996:
422:
of a right circular cylinder have been known from early antiquity.
4009:
2778:
2726:
1683:
697:
541:
between the secant plane and cylinder axis, in the following way:
494:
452:
165:
2228:{\displaystyle \rho =-H+{\frac {D^{2}}{4A}}+{\frac {E^{2}}{4B}}.}
2902:
2285:, then the equation of an elliptic cylinder may be rewritten in
1296:{\displaystyle A=L+2B=2\pi rh+2\pi r^{2}=2\pi r(h+r)=\pi d(r+h)}
3957:
3764:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999),
3614:
3612:
2827:
From a polyhedral viewpoint, a cylinder can also be seen as a
529:
of the cylindric section depend on the radius of the cylinder
1578:
The surface area, including the top and bottom, is given by
2811:. The connection is very strong and many older texts treat
2786:
building, Copenhagen, is an example of a truncated cylinder
2011:{\displaystyle f(x,y,z)=Ax^{2}+By^{2}+Cz^{2}+Dx+Ey+Gz+H=0,}
3540:, the intersection of two or three perpendicular cylinders
2410:
has a different sign than the coefficients, we obtain the
2383:, but that name is ambiguous, as it can also refer to the
1713:
by exploiting the relationship between a sphere and its
465:. They are, in general, curves and are special types of
425:
A right circular cylinder can also be thought of as the
250:
consisting of all the points on all the lines which are
3984:
Compact topological surfaces and their immersions in 3D
3786:
3784:
1728:
that of the circumscribed cylinder and a surface area
2686:
2574:
2544:
2502:
2420:
2396:
2295:
2263:
2159:
2048:
1893:
1845:
spanned by a one-parameter family of parallel lines.
1584:
1447:
1353:
1189:
1152:
931:
785:
653:
547:
3892:(Alternate ed.), Prindle, Weber & Schmidt,
475:. Such a cylindric section of a right cylinder is a
4196:
4168:
4133:
4124:
4070:
4025:
3996:
3989:
702:A solid elliptic right cylinder with the semi-axes
117:
105:
91:
81:
67:
39:
3885:
2717:
2636:
2556:
2514:
2488:
2402:
2357:
2269:
2227:
2145:
2010:
1666:
1551:
1402:Right circular hollow cylinder (cylindrical shell)
1374:
1295:
1173:
1056:
911:
678:
615:
156: 'roller, tumbler') has traditionally been a
309:of the cylinder. The two bases of a cylinder are
3851:(Rev. ed.), Allyn and Bacon, pp. 79–81
1827:In some areas of geometry and topology the term
3906:Wentworth, George; Smith, David Eugene (1913),
1559:Thus, the volume of a cylindrical shell equals
779:the area of each elliptic cross-section, thus:
499:Cylindric sections of a right circular cylinder
359:about a fixed line that it is parallel to is a
266:point of view, given a plane curve, called the
254:to a given line and which pass through a fixed
3703:
3661:
3630:
3618:
234:
3969:
3845:Solid Geometry with Problems and Applications
2867:
1848:A cylinder having a right section that is an
440:has a diameter much greater than its height.
8:
2771:, which may include the cylindrical conics.
2253:. Further simplification can be obtained by
191:in various modern branches of geometry and
4130:
3993:
3976:
3962:
3954:
3768:, Cambridge University Press, p. 34,
2874:
2860:
2841:
45:
3727:
3715:
3603:
2691:
2685:
2622:
2608:
2594:
2580:
2573:
2543:
2501:
2468:
2454:
2440:
2426:
2419:
2395:
2343:
2329:
2315:
2301:
2294:
2262:
2206:
2200:
2181:
2175:
2158:
2128:
2108:
2084:
2064:
2047:
1960:
1944:
1928:
1892:
1650:
1637:
1583:
1506:
1479:
1466:
1446:
1352:
1236:
1188:
1151:
1043:
1037:
1032:
1012:
1005:
998:
997:
988:
983:
970:
965:
955:
950:
932:
930:
879:
874:
837:
832:
801:
796:
784:
691:This formula may be established by using
667:
652:
630:If the base of a circular cylinder has a
588:
548:
546:
289:bounded by a cylindrical surface and two
2767:This concept is useful when considering
2379:). Elliptic cylinders are also known as
1878:
1392:
373:
281:A right and an oblique circular cylinder
276:
3549:
2857:
2680:with equations that can be written as:
2568:, whose equations may be rewritten as:
1127:. The area of the side is known as the
3802:
3790:
3751:
3739:
3584:, W. H. Freeman and Co., p. 607,
3566:, Henry George Liddell, Robert Scott,
2794:can be seen as the limiting case of a
2277:has the same sign as the coefficients
1698:In the treatise by this name, written
378:A right circular cylinder with radius
36:
1872:, respectively. These are degenerate
1785:. The surface area of this sphere is
183:A cylinder may also be defined as an
7:
2735:, a cylinder is simply a cone whose
2496:which have no real points on them. (
355:The cylinder obtained by rotating a
51:A circular right cylinder of height
752:is the area of the base ellipse (=
348:(or altitude) of a cylinder is the
262:of the cylindrical surface. From a
25:
1831:refers to what has been called a
3866:Albert, Abraham Adrian (2016) ,
3515:
3510:
3505:
3500:
3495:
3483:
3478:
3473:
3468:
3463:
3454:
3449:
3444:
3439:
3434:
3425:
3420:
3415:
3410:
3405:
3396:
3391:
3386:
3381:
3376:
3367:
3362:
3357:
3352:
3347:
3338:
3333:
3328:
3323:
3318:
3309:
3304:
3299:
3294:
3289:
3280:
3275:
3270:
3265:
3260:
3251:
3246:
3241:
3236:
3231:
3222:
3217:
3212:
3207:
3202:
3193:
3188:
3183:
3178:
3173:
3112:
3098:
3089:
3082:
3075:
3068:
3061:
3054:
3047:
3030:
3023:
3016:
3009:
3002:
2995:
2988:
2981:
2974:
2967:
2960:
1386:is the length of an element and
714:for the base ellipse and height
3888:Calculus with Analytic Geometry
3817:Geometry a Comprehensive Course
1317:of the circular top or bottom.
3677:; Terrell, Maria Shea (2013),
2257:and scalar multiplication. If
1915:
1897:
1543:
1531:
1408:right circular hollow cylinder
1290:
1278:
1266:
1254:
816:
810:
1:
1699:
1098:the area of the bottom base:
520:of the cylindric section and
329:(regions whose boundary is a
2718:{\displaystyle x^{2}+2ay=0.}
2522:gives a single real point.)
2412:imaginary elliptic cylinders
2018:with the coefficients being
1375:{\displaystyle L=e\times p,}
679:{\displaystyle V=\pi r^{2}h}
639:and the cylinder has height
352:distance between its bases.
319:, otherwise it is called an
3884:Swokowski, Earl W. (1983),
2557:{\displaystyle \rho \neq 0}
2249:this is the equation of an
333:) the cylinder is called a
160:, one of the most basic of
4280:
3942:Surface area of a cylinder
3704:Wentworth & Smith 1913
3680:Calculus With Applications
3662:Wentworth & Smith 1913
3631:Wentworth & Smith 1913
3619:Wentworth & Smith 1913
3580:Jacobs, Harold R. (1974),
2844:
2660:without loss of generality
1724:. The sphere has a volume
1694:On the Sphere and Cylinder
1691:
1679:On the Sphere and Cylinder
1441:. The volume is given by
1089:the area of the top base:
504:cylindrical surface in an
397:
235:Wentworth & Smith 1913
142:
29:
3683:, Springer, p. 178,
2752:, a cylinder is simply a
2538:have different signs and
1174:{\displaystyle L=2\pi rh}
44:
32:Cylinder (disambiguation)
4259:Euclidean solid geometry
3908:Plane and Solid Geometry
394:Right circular cylinders
233:and David Eugene Smith (
227:Plane and Solid Geometry
4115:Sphere with three holes
3868:Solid Analytic Geometry
3842:; Lennes, N.J. (1919),
3568:A Greek-English Lexicon
3044:Spherical tiling image
2792:solid circular cylinder
2784:Tycho Brahe Planetarium
2515:{\displaystyle \rho =0}
1720:of the same height and
1718:right circular cylinder
923:cylindrical coordinates
414:. The formulae for the
400:Right circular cylinder
301:element of the cylinder
214:right circular cylinder
158:three-dimensional solid
27:Three-dimensional solid
3819:, Dover, p. 398,
2787:
2740:
2719:
2638:
2558:
2516:
2490:
2404:
2359:
2271:
2229:
2147:
2012:
1884:
1689:
1668:
1569:average radius ×
1553:
1435:, and external radius
1398:
1376:
1297:
1175:
1107:the area of the side:
1077:and altitude (height)
1058:
913:
720:
680:
617:
500:
458:
390:
362:cylinder of revolution
282:
4033:Real projective plane
4018:Pretzel (genus 3) ...
2835:as an infinite-sided
2782:
2760:(vertex) lies on the
2730:
2720:
2639:
2559:
2517:
2491:
2405:
2403:{\displaystyle \rho }
2360:
2287:Cartesian coordinates
2272:
2270:{\displaystyle \rho }
2230:
2148:
2013:
1882:
1838:generalized cylinders
1687:
1669:
1554:
1396:
1377:
1298:
1176:
1059:
914:
701:
693:Cavalieri's principle
681:
618:
498:
456:
377:
280:
172:, it is considered a
4188:Euler characteristic
3948:Volume of a cylinder
3815:Pedoe, Dan (1988) ,
2684:
2572:
2566:hyperbolic cylinders
2542:
2500:
2418:
2394:
2293:
2261:
2157:
2046:
1891:
1823:Cylindrical surfaces
1582:
1445:
1351:
1187:
1150:
929:
783:
651:
545:
293:is called a (solid)
209:cylindrical surfaces
30:For other uses, see
3644:"Cylindric section"
3109:Plane tiling image
2750:projective geometry
2744:Projective geometry
2733:projective geometry
2678:parabolic cylinders
2526:Hyperbolic cylinder
2255:translation of axes
1870:hyperbolic cylinder
1833:cylindrical surface
993:
978:
960:
884:
842:
806:
427:solid of revolution
325:. If the bases are
243:cylindrical surface
231:George A. Wentworth
170:elementary geometry
4015:Number 8 (genus 2)
3925:Weisstein, Eric W.
3562:2013-07-30 at the
2938:Hendecagonal prism
2822:truncated cylinder
2788:
2741:
2715:
2647:Parabolic cylinder
2634:
2554:
2512:
2486:
2400:
2355:
2267:
2225:
2143:
2038:that the variable
2008:
1885:
1883:Parabolic cylinder
1866:parabolic cylinder
1690:
1664:
1575: thickness.
1549:
1429:, internal radius
1423:Let the height be
1399:
1372:
1343:The lateral area,
1293:
1171:
1054:
1052:
979:
961:
946:
909:
870:
828:
792:
730:, semi-minor axis
721:
676:
613:
611:
501:
459:
449:Cylindric sections
391:
283:
4254:Elementary shapes
4236:
4235:
4232:
4231:
4066:
4065:
3877:978-0-486-81026-3
3775:978-0-521-59787-6
3524:
3523:
2957:Polyhedron image
2951:Apeirogonal prism
2943:Dodecagonal prism
2769:degenerate conics
2762:plane at infinity
2616:
2588:
2462:
2434:
2367:circular cylinder
2337:
2309:
2251:elliptic cylinder
2238:Elliptic cylinder
2220:
2195:
2121:
2077:
1862:elliptic cylinder
1522:
1413:cylindrical shell
604:
457:Cylindric section
336:circular cylinder
129:
128:
76:Algebraic surface
16:(Redirected from
4271:
4151:Triangulatedness
4131:
3994:
3990:Without boundary
3978:
3971:
3964:
3955:
3938:
3937:
3911:
3902:
3891:
3880:
3853:
3852:
3850:
3836:
3830:
3829:
3812:
3806:
3800:
3794:
3788:
3779:
3778:
3761:
3755:
3749:
3743:
3737:
3731:
3725:
3719:
3713:
3707:
3701:
3695:
3693:
3671:
3665:
3659:
3653:
3652:
3640:
3634:
3628:
3622:
3616:
3607:
3601:
3595:
3594:
3577:
3571:
3554:
3520:
3519:
3518:
3514:
3513:
3509:
3508:
3504:
3503:
3499:
3498:
3488:
3487:
3486:
3482:
3481:
3477:
3476:
3472:
3471:
3467:
3466:
3459:
3458:
3457:
3453:
3452:
3448:
3447:
3443:
3442:
3438:
3437:
3430:
3429:
3428:
3424:
3423:
3419:
3418:
3414:
3413:
3409:
3408:
3401:
3400:
3399:
3395:
3394:
3390:
3389:
3385:
3384:
3380:
3379:
3372:
3371:
3370:
3366:
3365:
3361:
3360:
3356:
3355:
3351:
3350:
3343:
3342:
3341:
3337:
3336:
3332:
3331:
3327:
3326:
3322:
3321:
3314:
3313:
3312:
3308:
3307:
3303:
3302:
3298:
3297:
3293:
3292:
3285:
3284:
3283:
3279:
3278:
3274:
3273:
3269:
3268:
3264:
3263:
3256:
3255:
3254:
3250:
3249:
3245:
3244:
3240:
3239:
3235:
3234:
3227:
3226:
3225:
3221:
3220:
3216:
3215:
3211:
3210:
3206:
3205:
3198:
3197:
3196:
3192:
3191:
3187:
3186:
3182:
3181:
3177:
3176:
3116:
3102:
3093:
3086:
3079:
3072:
3065:
3058:
3051:
3034:
3027:
3020:
3013:
3006:
2999:
2992:
2985:
2978:
2971:
2964:
2928:Enneagonal prism
2918:Heptagonal prism
2908:Pentagonal prism
2896:Triangular prism
2876:
2869:
2862:
2842:
2806:
2798:
2724:
2722:
2721:
2716:
2696:
2695:
2675:
2668:
2657:
2643:
2641:
2640:
2635:
2627:
2626:
2621:
2617:
2609:
2599:
2598:
2593:
2589:
2581:
2564:, we obtain the
2563:
2561:
2560:
2555:
2537:
2533:
2521:
2519:
2518:
2513:
2495:
2493:
2492:
2487:
2473:
2472:
2467:
2463:
2455:
2445:
2444:
2439:
2435:
2427:
2409:
2407:
2406:
2401:
2378:
2364:
2362:
2361:
2356:
2348:
2347:
2342:
2338:
2330:
2320:
2319:
2314:
2310:
2302:
2284:
2280:
2276:
2274:
2273:
2268:
2248:
2234:
2232:
2231:
2226:
2221:
2219:
2211:
2210:
2201:
2196:
2194:
2186:
2185:
2176:
2152:
2150:
2149:
2144:
2133:
2132:
2127:
2123:
2122:
2120:
2109:
2089:
2088:
2083:
2079:
2078:
2076:
2065:
2041:
2036:rotation of axes
2033:
2029:
2025:
2017:
2015:
2014:
2009:
1965:
1964:
1949:
1948:
1933:
1932:
1874:quadric surfaces
1818:
1813:
1809:
1807:
1806:
1803:
1800:
1790:
1784:
1779:
1775:
1773:
1772:
1769:
1766:
1756:
1753:
1751:
1750:
1747:
1744:
1736:
1731:
1727:
1704:
1701:
1673:
1671:
1670:
1665:
1660:
1656:
1655:
1654:
1642:
1641:
1615:
1611:
1574:
1573:altitude ×
1570:
1566:
1564:
1558:
1556:
1555:
1550:
1527:
1523:
1518:
1507:
1489:
1485:
1484:
1483:
1471:
1470:
1440:
1434:
1428:
1389:
1385:
1381:
1379:
1378:
1373:
1346:
1339:
1329:
1312:
1302:
1300:
1299:
1294:
1241:
1240:
1180:
1178:
1177:
1172:
1138:
1126:
1113:
1104:
1095:
1080:
1076:
1063:
1061:
1060:
1055:
1053:
1042:
1041:
1022:
992:
987:
977:
969:
959:
954:
918:
916:
915:
910:
883:
878:
841:
836:
805:
800:
778:
764:
760:
756:
751:
747:
737:
733:
729:
719:
713:
707:
685:
683:
682:
677:
672:
671:
642:
638:
622:
620:
619:
614:
612:
605:
603:
589:
540:
534:
528:
519:
432:A tall and thin
389:
383:
322:oblique cylinder
153:
146:
125:
113:
100:
49:
37:
21:
4279:
4278:
4274:
4273:
4272:
4270:
4269:
4268:
4239:
4238:
4237:
4228:
4192:
4169:Characteristics
4164:
4126:
4120:
4062:
4021:
3985:
3982:
3923:
3922:
3919:
3914:
3905:
3900:
3883:
3878:
3865:
3861:
3856:
3848:
3838:
3837:
3833:
3827:
3814:
3813:
3809:
3801:
3797:
3789:
3782:
3776:
3763:
3762:
3758:
3750:
3746:
3738:
3734:
3726:
3722:
3714:
3710:
3702:
3698:
3691:
3673:
3672:
3668:
3660:
3656:
3642:
3641:
3637:
3629:
3625:
3617:
3610:
3602:
3598:
3592:
3579:
3578:
3574:
3564:Wayback Machine
3555:
3551:
3547:
3538:Steinmetz solid
3529:
3516:
3511:
3506:
3501:
3496:
3494:
3484:
3479:
3474:
3469:
3464:
3462:
3455:
3450:
3445:
3440:
3435:
3433:
3426:
3421:
3416:
3411:
3406:
3404:
3397:
3392:
3387:
3382:
3377:
3375:
3368:
3363:
3358:
3353:
3348:
3346:
3339:
3334:
3329:
3324:
3319:
3317:
3310:
3305:
3300:
3295:
3290:
3288:
3281:
3276:
3271:
3266:
3261:
3259:
3252:
3247:
3242:
3237:
3232:
3230:
3223:
3218:
3213:
3208:
3203:
3201:
3194:
3189:
3184:
3179:
3174:
3172:
3168:Coxeter diagram
2933:Decagonal prism
2923:Octagonal prism
2913:Hexagonal prism
2901:
2894:
2880:
2818:truncated prism
2802:
2796:
2777:
2746:
2687:
2682:
2681:
2670:
2663:
2652:
2649:
2604:
2603:
2576:
2575:
2570:
2569:
2540:
2539:
2535:
2531:
2528:
2498:
2497:
2450:
2449:
2422:
2421:
2416:
2415:
2392:
2391:
2370:
2325:
2324:
2297:
2296:
2291:
2290:
2282:
2278:
2259:
2258:
2243:
2240:
2212:
2202:
2187:
2177:
2155:
2154:
2113:
2101:
2097:
2096:
2069:
2057:
2053:
2052:
2044:
2043:
2039:
2031:
2027:
2023:
2022:and not all of
1956:
1940:
1924:
1889:
1888:
1839:
1825:
1811:
1804:
1801:
1798:
1797:
1795:
1788:
1786:
1777:
1770:
1767:
1764:
1763:
1761:
1754:
1748:
1745:
1742:
1741:
1739:
1738:
1734:
1729:
1725:
1702:
1696:
1682:
1646:
1633:
1632:
1628:
1601:
1597:
1580:
1579:
1572:
1568:
1562:
1560:
1508:
1502:
1475:
1462:
1461:
1457:
1443:
1442:
1436:
1430:
1424:
1414:
1404:
1397:Hollow cylinder
1387:
1383:
1349:
1348:
1344:
1331:
1321:
1304:
1232:
1185:
1184:
1148:
1147:
1134:
1131:
1122:
1108:
1099:
1090:
1078:
1072:
1069:
1051:
1050:
1033:
1020:
1019:
939:
927:
926:
781:
780:
766:
762:
754:
753:
749:
739:
735:
731:
727:
725:semi-major axis
715:
709:
703:
663:
649:
648:
640:
634:
628:
610:
609:
593:
581:
575:
574:
555:
543:
542:
536:
530:
524:
522:semi-major axis
515:
486:
451:
446:
434:needle cylinder
412:
402:
396:
385:
379:
369:
363:
346:
337:
323:
317:
307:
297:
291:parallel planes
244:
223:
205:solid cylinders
123:
111:
98:
74:
63:
35:
28:
23:
22:
15:
12:
11:
5:
4277:
4275:
4267:
4266:
4261:
4256:
4251:
4241:
4240:
4234:
4233:
4230:
4229:
4227:
4226:
4221:
4215:
4209:
4206:
4200:
4198:
4194:
4193:
4191:
4190:
4185:
4180:
4172:
4170:
4166:
4165:
4163:
4162:
4157:
4148:
4143:
4137:
4135:
4128:
4122:
4121:
4119:
4118:
4112:
4111:
4110:
4100:
4099:
4098:
4093:
4085:
4084:
4083:
4074:
4072:
4068:
4067:
4064:
4063:
4061:
4060:
4057:Dyck's surface
4054:
4048:
4047:
4046:
4041:
4029:
4027:
4026:Non-orientable
4023:
4022:
4020:
4019:
4016:
4013:
4007:
4000:
3998:
3991:
3987:
3986:
3983:
3981:
3980:
3973:
3966:
3958:
3952:
3951:
3945:
3939:
3918:
3917:External links
3915:
3913:
3912:
3910:, Ginn and Co.
3903:
3898:
3881:
3876:
3862:
3860:
3857:
3855:
3854:
3831:
3825:
3807:
3795:
3780:
3774:
3756:
3744:
3732:
3730:, p. 291.
3728:Swokowski 1983
3720:
3718:, p. 292.
3716:Swokowski 1983
3708:
3706:, p. 358.
3696:
3689:
3666:
3664:, p. 359.
3654:
3635:
3633:, p. 357.
3623:
3621:, p. 354.
3608:
3606:, p. 283.
3604:Swokowski 1983
3596:
3590:
3572:
3548:
3546:
3543:
3542:
3541:
3535:
3533:List of shapes
3528:
3525:
3522:
3521:
3492:
3489:
3460:
3431:
3402:
3373:
3344:
3315:
3286:
3257:
3228:
3199:
3170:
3164:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3122:Vertex config.
3118:
3117:
3110:
3107:
3105:
3103:
3096:
3094:
3087:
3080:
3073:
3066:
3059:
3052:
3045:
3041:
3040:
3038:
3035:
3028:
3021:
3014:
3007:
3000:
2993:
2986:
2979:
2972:
2965:
2958:
2954:
2953:
2948:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2898:
2891:
2886:
2882:
2881:
2879:
2878:
2871:
2864:
2856:
2776:
2773:
2745:
2742:
2714:
2711:
2708:
2705:
2702:
2699:
2694:
2690:
2676:to obtain the
2648:
2645:
2633:
2630:
2625:
2620:
2615:
2612:
2607:
2602:
2597:
2592:
2587:
2584:
2579:
2553:
2550:
2547:
2527:
2524:
2511:
2508:
2505:
2485:
2482:
2479:
2476:
2471:
2466:
2461:
2458:
2453:
2448:
2443:
2438:
2433:
2430:
2425:
2399:
2385:Plücker conoid
2354:
2351:
2346:
2341:
2336:
2333:
2328:
2323:
2318:
2313:
2308:
2305:
2300:
2266:
2239:
2236:
2224:
2218:
2215:
2209:
2205:
2199:
2193:
2190:
2184:
2180:
2174:
2171:
2168:
2165:
2162:
2142:
2139:
2136:
2131:
2126:
2119:
2116:
2112:
2107:
2104:
2100:
2095:
2092:
2087:
2082:
2075:
2072:
2068:
2063:
2060:
2056:
2051:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1963:
1959:
1955:
1952:
1947:
1943:
1939:
1936:
1931:
1927:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1837:
1824:
1821:
1703: 225 BCE
1692:Main article:
1681:
1676:
1663:
1659:
1653:
1649:
1645:
1640:
1636:
1631:
1627:
1624:
1621:
1618:
1614:
1610:
1607:
1604:
1600:
1596:
1593:
1590:
1587:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1526:
1521:
1517:
1514:
1511:
1505:
1501:
1498:
1495:
1492:
1488:
1482:
1478:
1474:
1469:
1465:
1460:
1456:
1453:
1450:
1412:
1403:
1400:
1371:
1368:
1365:
1362:
1359:
1356:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1239:
1235:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1170:
1167:
1164:
1161:
1158:
1155:
1129:
1115:
1114:
1105:
1096:
1071:Having radius
1068:
1065:
1049:
1046:
1040:
1036:
1031:
1028:
1025:
1023:
1021:
1018:
1015:
1011:
1008:
1004:
1001:
996:
991:
986:
982:
976:
973:
968:
964:
958:
953:
949:
945:
942:
940:
938:
935:
934:
908:
905:
902:
899:
896:
893:
890:
887:
882:
877:
873:
869:
866:
863:
860:
857:
854:
851:
848:
845:
840:
835:
831:
827:
824:
821:
818:
815:
812:
809:
804:
799:
795:
791:
788:
675:
670:
666:
662:
659:
656:
627:
624:
608:
602:
599:
596:
592:
587:
584:
582:
580:
577:
576:
573:
570:
567:
564:
561:
558:
556:
554:
551:
550:
535:and the angle
484:
468:plane sections
450:
447:
445:
442:
410:
404:The bare term
398:Main article:
395:
392:
367:
361:
344:
335:
321:
316:right cylinder
315:
305:
295:
242:
222:
219:
127:
126:
121:
115:
114:
109:
103:
102:
95:
93:Symmetry group
89:
88:
85:
79:
78:
72:Smooth surface
69:
65:
64:
50:
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4276:
4265:
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4246:
4244:
4225:
4222:
4220:
4216:
4214:
4210:
4208:Making a hole
4207:
4205:
4204:Connected sum
4202:
4201:
4199:
4195:
4189:
4186:
4184:
4181:
4178:
4174:
4173:
4171:
4167:
4161:
4160:Orientability
4158:
4156:
4152:
4149:
4147:
4144:
4142:
4141:Connectedness
4139:
4138:
4136:
4132:
4129:
4123:
4116:
4113:
4109:
4106:
4105:
4104:
4101:
4097:
4094:
4092:
4089:
4088:
4086:
4081:
4080:
4079:
4076:
4075:
4073:
4071:With boundary
4069:
4059:(genus 3) ...
4058:
4055:
4052:
4049:
4045:
4044:Roman surface
4042:
4040:
4039:Boy's surface
4036:
4035:
4034:
4031:
4030:
4028:
4024:
4017:
4014:
4011:
4008:
4005:
4002:
4001:
3999:
3995:
3992:
3988:
3979:
3974:
3972:
3967:
3965:
3960:
3959:
3956:
3949:
3946:
3943:
3940:
3935:
3934:
3929:
3926:
3921:
3920:
3916:
3909:
3904:
3901:
3899:0-87150-341-7
3895:
3890:
3889:
3882:
3879:
3873:
3869:
3864:
3863:
3858:
3847:
3846:
3841:
3840:Slaught, H.E.
3835:
3832:
3828:
3826:0-486-65812-0
3822:
3818:
3811:
3808:
3805:, p. 75.
3804:
3799:
3796:
3793:, p. 74.
3792:
3787:
3785:
3781:
3777:
3771:
3767:
3760:
3757:
3754:, p. 49.
3753:
3748:
3745:
3742:, p. 43.
3741:
3736:
3733:
3729:
3724:
3721:
3717:
3712:
3709:
3705:
3700:
3697:
3692:
3690:9781461479468
3686:
3682:
3681:
3676:
3675:Lax, Peter D.
3670:
3667:
3663:
3658:
3655:
3651:
3650:
3645:
3639:
3636:
3632:
3627:
3624:
3620:
3615:
3613:
3609:
3605:
3600:
3597:
3593:
3591:0-7167-0456-0
3587:
3583:
3576:
3573:
3569:
3565:
3561:
3558:
3553:
3550:
3544:
3539:
3536:
3534:
3531:
3530:
3526:
3493:
3490:
3461:
3432:
3403:
3374:
3345:
3316:
3287:
3258:
3229:
3200:
3171:
3169:
3166:
3165:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3123:
3120:
3119:
3115:
3111:
3108:
3106:
3104:
3101:
3097:
3095:
3092:
3088:
3085:
3081:
3078:
3074:
3071:
3067:
3064:
3060:
3057:
3053:
3050:
3046:
3043:
3042:
3039:
3036:
3033:
3029:
3026:
3022:
3019:
3015:
3012:
3008:
3005:
3001:
2998:
2994:
2991:
2987:
2984:
2980:
2977:
2973:
2970:
2966:
2963:
2959:
2956:
2955:
2952:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2899:
2897:
2892:
2890:
2889:Digonal prism
2887:
2884:
2883:
2877:
2872:
2870:
2865:
2863:
2858:
2855:
2851:
2848:
2843:
2840:
2838:
2834:
2830:
2825:
2823:
2819:
2814:
2810:
2805:
2800:
2793:
2785:
2781:
2774:
2772:
2770:
2765:
2763:
2759:
2755:
2751:
2743:
2738:
2734:
2729:
2725:
2712:
2709:
2706:
2703:
2700:
2697:
2692:
2688:
2679:
2673:
2666:
2661:
2655:
2646:
2644:
2631:
2628:
2623:
2618:
2613:
2610:
2605:
2600:
2595:
2590:
2585:
2582:
2577:
2567:
2551:
2548:
2545:
2525:
2523:
2509:
2506:
2503:
2483:
2480:
2477:
2474:
2469:
2464:
2459:
2456:
2451:
2446:
2441:
2436:
2431:
2428:
2423:
2413:
2397:
2388:
2386:
2382:
2377:
2373:
2368:
2352:
2349:
2344:
2339:
2334:
2331:
2326:
2321:
2316:
2311:
2306:
2303:
2298:
2288:
2264:
2256:
2252:
2246:
2237:
2235:
2222:
2216:
2213:
2207:
2203:
2197:
2191:
2188:
2182:
2178:
2172:
2169:
2166:
2163:
2160:
2140:
2137:
2134:
2129:
2124:
2117:
2114:
2110:
2105:
2102:
2098:
2093:
2090:
2085:
2080:
2073:
2070:
2066:
2061:
2058:
2054:
2049:
2037:
2021:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1961:
1957:
1953:
1950:
1945:
1941:
1937:
1934:
1929:
1925:
1921:
1918:
1912:
1909:
1906:
1903:
1900:
1894:
1881:
1877:
1875:
1871:
1867:
1863:
1860:is called an
1859:
1855:
1851:
1846:
1844:
1843:ruled surface
1840:
1834:
1830:
1822:
1820:
1816:
1793:
1782:
1759:
1723:
1719:
1716:
1715:circumscribed
1712:
1708:
1695:
1686:
1680:
1677:
1675:
1661:
1657:
1651:
1647:
1643:
1638:
1634:
1629:
1625:
1622:
1619:
1616:
1612:
1608:
1605:
1602:
1598:
1594:
1591:
1588:
1585:
1576:
1546:
1540:
1537:
1534:
1528:
1524:
1519:
1515:
1512:
1509:
1503:
1499:
1496:
1493:
1490:
1486:
1480:
1476:
1472:
1467:
1463:
1458:
1454:
1451:
1448:
1439:
1433:
1427:
1421:
1419:
1415:
1409:
1401:
1395:
1391:
1369:
1366:
1363:
1360:
1357:
1354:
1341:
1338:
1334:
1328:
1324:
1318:
1316:
1311:
1307:
1287:
1284:
1281:
1275:
1272:
1269:
1263:
1260:
1257:
1251:
1248:
1245:
1242:
1237:
1233:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1181:
1168:
1165:
1162:
1159:
1156:
1153:
1145:
1144:open cylinder
1140:
1137:
1132:
1125:
1120:
1112:
1106:
1103:
1097:
1094:
1088:
1087:
1086:
1084:
1075:
1066:
1064:
1047:
1044:
1038:
1034:
1029:
1026:
1024:
1016:
1013:
1009:
1006:
1002:
999:
994:
989:
984:
980:
974:
971:
966:
962:
956:
951:
947:
943:
941:
936:
924:
919:
906:
903:
900:
897:
894:
891:
888:
885:
880:
875:
871:
867:
864:
861:
858:
855:
852:
849:
846:
843:
838:
833:
829:
825:
822:
819:
813:
807:
802:
797:
793:
789:
786:
777:
773:
769:
759:
746:
742:
738:has a volume
726:
718:
712:
706:
700:
696:
694:
689:
686:
673:
668:
664:
660:
657:
654:
646:
637:
633:
625:
623:
606:
600:
597:
594:
590:
585:
583:
578:
571:
568:
565:
562:
559:
557:
552:
539:
533:
527:
523:
518:
514:
509:
507:
497:
493:
491:
490:conic section
487:
485:right section
480:
478:
474:
473:parallelogram
470:
469:
464:
455:
448:
443:
441:
439:
438:disk cylinder
435:
430:
428:
423:
421:
417:
413:
411:open cylinder
407:
401:
393:
388:
382:
376:
372:
370:
364:
358:
353:
351:
350:perpendicular
347:
340:
338:
332:
328:
324:
318:
312:
308:
302:
298:
292:
288:
279:
275:
273:
269:
265:
261:
257:
253:
249:
245:
238:
236:
232:
228:
220:
218:
216:
215:
210:
206:
202:
198:
194:
190:
186:
181:
180:as its base.
179:
175:
171:
167:
163:
159:
155:
152:
145:
141:
138:
137:Ancient Greek
134:
122:
120:
116:
110:
108:
104:
101:
96:
94:
90:
86:
84:
80:
77:
73:
70:
66:
62:
58:
55:and diameter
54:
48:
43:
38:
33:
19:
4103:Möbius strip
4095:
4051:Klein bottle
3950:at MATHguide
3944:at MATHguide
3931:
3907:
3887:
3867:
3844:
3834:
3816:
3810:
3798:
3765:
3759:
3747:
3735:
3723:
3711:
3699:
3679:
3669:
3657:
3647:
3638:
3626:
3599:
3581:
3575:
3570:, on Perseus
3567:
3552:
2903:Square prism
2900:(Tetragonal)
2849:
2826:
2821:
2817:
2803:
2801:prism where
2791:
2789:
2766:
2747:
2677:
2671:
2664:
2653:
2651:Finally, if
2650:
2565:
2529:
2411:
2389:
2380:
2375:
2371:
2366:
2250:
2244:
2241:
2020:real numbers
1886:
1869:
1865:
1861:
1847:
1836:
1832:
1828:
1826:
1814:
1791:
1780:
1757:
1697:
1678:
1577:
1437:
1431:
1425:
1422:
1411:
1407:
1405:
1342:
1336:
1332:
1326:
1322:
1319:
1309:
1305:
1182:
1143:
1141:
1135:
1130:lateral area
1128:
1123:
1118:
1116:
1110:
1101:
1092:
1083:surface area
1073:
1070:
1067:Surface area
920:
775:
771:
767:
757:
744:
740:
722:
716:
710:
704:
690:
687:
647:is given by
635:
629:
537:
531:
525:
516:
513:eccentricity
510:
502:
483:
481:
467:
460:
437:
433:
431:
424:
416:surface area
409:
405:
403:
386:
380:
366:
360:
357:line segment
354:
343:
341:
334:
320:
314:
304:
300:
294:
284:
271:
267:
259:
241:
239:
226:
224:
212:
208:
204:
187:curvilinear
182:
150:
147:
140:
132:
130:
107:Surface area
60:
56:
52:
4146:Compactness
3803:Albert 2016
3791:Albert 2016
3752:Albert 2016
3740:Albert 2016
2885:Prism name
2807:approaches
2381:cylindroids
734:and height
643:, then its
384:and height
256:plane curve
162:curvilinear
83:Euler char.
18:Cylindrical
4243:Categories
4197:Operations
4179:components
4175:Number of
4155:smoothness
4134:Properties
4082:Semisphere
3997:Orientable
3928:"Cylinder"
3859:References
2893:(Trigonal)
2845:Family of
1730:two-thirds
1726:two-thirds
1707:Archimedes
765:-axis and
444:Properties
272:generatrix
264:kinematics
164:geometric
135:(from
112:2πr(r + h)
4224:Immersion
4219:cross-cap
4217:Gluing a
4211:Gluing a
4108:Cross-cap
4053:(genus 2)
4037:genus 1;
4012:(genus 1)
4006:(genus 0)
3933:MathWorld
3870:, Dover,
3649:MathWorld
3557:κύλινδρος
2837:bipyramid
2601:−
2549:≠
2546:ρ
2504:ρ
2478:−
2398:ρ
2265:ρ
2167:−
2161:ρ
2138:ρ
1858:hyperbola
1644:−
1626:π
1595:π
1538:−
1500:π
1473:−
1455:π
1364:×
1273:π
1249:π
1230:π
1215:π
1163:π
1119:base area
1030:π
1010:ϕ
981:∫
975:π
963:∫
948:∫
895:π
872:∫
862:π
844:π
830:∫
794:∫
661:π
601:α
598:
569:α
566:
477:rectangle
311:congruent
268:directrix
151:kúlindros
144:κύλινδρος
99:O(2)×O(1)
4264:Surfaces
4249:Quadrics
4177:boundary
4096:Cylinder
3766:Geometry
3582:Geometry
3560:Archived
3527:See also
2809:infinity
2658:assume,
1854:parabola
1829:cylinder
1722:diameter
1565: ×
1315:diameter
748:, where
418:and the
406:cylinder
296:cylinder
252:parallel
193:topology
185:infinite
133:cylinder
40:Cylinder
4127:notions
4125:Related
4091:Annulus
4087:Ribbon
2852:-gonal
2847:uniform
2662:, that
1850:ellipse
1808:
1796:
1774:
1762:
1752:
1740:
1571:
1567:
1418:annular
1313:is the
506:ellipse
260:element
248:surface
199:versus
189:surface
176:with a
4213:handle
4004:Sphere
3896:
3874:
3823:
3772:
3687:
3588:
3162:∞.4.4
3156:12.4.4
3153:11.4.4
3150:10.4.4
2854:prisms
2833:bicone
2813:prisms
2799:-gonal
2775:Prisms
2756:whose
2247:> 0
2153:where
1711:sphere
1382:where
1303:where
1081:, the
921:Using
645:volume
632:radius
626:Volume
420:volume
345:height
331:circle
201:sphere
178:circle
166:shapes
119:Volume
4183:Genus
4010:Torus
3849:(PDF)
3545:Notes
3147:9.4.4
3144:8.4.4
3141:7.4.4
3138:6.4.4
3135:5.4.4
3132:4.4.4
3129:3.4.4
3126:2.4.4
2831:of a
1856:, or
463:plane
327:disks
287:solid
246:is a
221:Types
174:prism
168:. In
139:
4078:Disk
3894:ISBN
3872:ISBN
3821:ISBN
3770:ISBN
3685:ISBN
3586:ISBN
3491:...
3037:...
2947:...
2829:dual
2758:apex
2754:cone
2737:apex
2669:and
2534:and
2289:as:
2281:and
2030:and
1868:and
1410:(or
774:) =
708:and
368:axis
342:The
306:base
207:and
197:ball
68:Type
4153:or
4117:...
3159:...
2748:In
2731:In
2674:= 1
2667:= 0
2656:= 0
2530:If
2390:If
2242:If
1737:is
1335:= 2
1325:= 2
1308:= 2
1142:An
595:sin
563:cos
237:).
229:by
124:πrh
4245::
3930:.
3783:^
3646:,
3611:^
2839:.
2824:.
2790:A
2713:0.
2654:AB
2632:1.
2414::
2387:.
2374:=
2353:1.
2245:AB
2026:,
1876:.
1864:,
1852:,
1810:(6
1794:=
1776:(2
1760:=
1705:,
1700:c.
1406:A
1139:.
1133:,
1121:,
1111:rh
1109:2π
758:ab
745:Ah
743:=
695:.
479:.
285:A
240:A
217:.
131:A
59:=2
3977:e
3970:t
3963:v
3936:.
3694:.
2875:e
2868:t
2861:v
2850:n
2804:n
2797:n
2710:=
2707:y
2704:a
2701:2
2698:+
2693:2
2689:x
2672:A
2665:B
2629:=
2624:2
2619:)
2614:b
2611:y
2606:(
2596:2
2591:)
2586:a
2583:x
2578:(
2552:0
2536:B
2532:A
2510:0
2507:=
2484:,
2481:1
2475:=
2470:2
2465:)
2460:b
2457:y
2452:(
2447:+
2442:2
2437:)
2432:a
2429:x
2424:(
2376:b
2372:a
2369:(
2350:=
2345:2
2340:)
2335:b
2332:y
2327:(
2322:+
2317:2
2312:)
2307:a
2304:x
2299:(
2283:B
2279:A
2223:.
2217:B
2214:4
2208:2
2204:E
2198:+
2192:A
2189:4
2183:2
2179:D
2173:+
2170:H
2164:=
2141:,
2135:=
2130:2
2125:)
2118:B
2115:2
2111:E
2106:+
2103:y
2099:(
2094:B
2091:+
2086:2
2081:)
2074:A
2071:2
2067:D
2062:+
2059:x
2055:(
2050:A
2040:z
2032:C
2028:B
2024:A
2006:,
2003:0
2000:=
1997:H
1994:+
1991:z
1988:G
1985:+
1982:y
1979:E
1976:+
1973:x
1970:D
1967:+
1962:2
1958:z
1954:C
1951:+
1946:2
1942:y
1938:B
1935:+
1930:2
1926:x
1922:A
1919:=
1916:)
1913:z
1910:,
1907:y
1904:,
1901:x
1898:(
1895:f
1817:)
1815:r
1812:π
1805:3
1802:/
1799:2
1792:r
1789:π
1787:4
1783:)
1781:r
1778:π
1771:3
1768:/
1765:2
1758:r
1755:π
1749:3
1746:/
1743:4
1735:r
1662:.
1658:)
1652:2
1648:r
1639:2
1635:R
1630:(
1623:2
1620:+
1617:h
1613:)
1609:r
1606:+
1603:R
1599:(
1592:2
1589:=
1586:A
1563:π
1561:2
1547:.
1544:)
1541:r
1535:R
1532:(
1529:h
1525:)
1520:2
1516:r
1513:+
1510:R
1504:(
1497:2
1494:=
1491:h
1487:)
1481:2
1477:r
1468:2
1464:R
1459:(
1452:=
1449:V
1438:R
1432:r
1426:h
1388:p
1384:e
1370:,
1367:p
1361:e
1358:=
1355:L
1345:L
1337:r
1333:h
1327:r
1323:h
1310:r
1306:d
1291:)
1288:h
1285:+
1282:r
1279:(
1276:d
1270:=
1267:)
1264:r
1261:+
1258:h
1255:(
1252:r
1246:2
1243:=
1238:2
1234:r
1227:2
1224:+
1221:h
1218:r
1212:2
1209:=
1206:B
1203:2
1200:+
1197:L
1194:=
1191:A
1169:h
1166:r
1160:2
1157:=
1154:L
1136:L
1124:B
1102:r
1100:π
1093:r
1091:π
1079:h
1074:r
1048:.
1045:h
1039:2
1035:r
1027:=
1017:z
1014:d
1007:d
1003:s
1000:d
995:s
990:r
985:0
972:2
967:0
957:h
952:0
944:=
937:V
907:.
904:h
901:b
898:a
892:=
889:x
886:d
881:h
876:0
868:b
865:a
859:=
856:x
853:d
850:b
847:a
839:h
834:0
826:=
823:x
820:d
817:)
814:x
811:(
808:A
803:h
798:0
790:=
787:V
776:A
772:x
770:(
768:A
763:x
755:π
750:A
741:V
736:h
732:b
728:a
717:h
711:b
705:a
674:h
669:2
665:r
658:=
655:V
641:h
636:r
607:.
591:r
586:=
579:a
572:,
560:=
553:e
538:α
532:r
526:a
517:e
387:h
381:r
154:)
148:(
87:2
61:r
57:d
53:h
34:.
20:)
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