2717:
2769:
3073:
3059:
3045:
3089:
3080:
3066:
1674:
3038:
3052:
688:
364:
1869:
443:
3014:
1383:
485:
3021:
3103:
497:. 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.
2972:
2965:
2958:
3007:
2993:
2979:
267:
36:
2986:
3000:
2805:
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
2753:. 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.
2804:
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
492:
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
1051:
1824:. 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
610:
2140:
906:
1546:
2483:
2631:
2352:
917:
1661:
2222:
1290:
1721:
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
922:
538:
2005:
354:. 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
397:
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
263:, 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.
533:
2034:
1830:. 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
418:
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.
771:
712:
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
3548:
1433:
2951:
1876:
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
3506:
3486:
3474:
3454:
3445:
3425:
3416:
3396:
3387:
3367:
3358:
3338:
3329:
3309:
3300:
3280:
3271:
3251:
3242:
3222:
3213:
3193:
3184:
3164:
2712:
1369:
673:
3491:
2551:
3496:
3464:
3435:
3406:
3401:
3377:
3348:
3319:
3290:
3261:
3232:
3203:
3174:
2406:
2560:
2281:
1168:
3459:
3430:
3501:
3469:
3440:
3411:
3382:
3372:
3353:
3343:
3324:
3314:
3295:
3285:
3266:
3256:
3237:
3227:
3208:
3198:
3179:
3169:
2509:
2397:
2264:
3964:
2023:
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
1570:
1379:
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.
2145:
1175:
1172:
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
477:. 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
1879:
292:. 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
1046:{\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}}}
3930:
750:). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive
132:
3957:
3545:
3936:
471:
A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a
2809:
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
2862:
3950:
3864:
3762:
200:. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the
481:(parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.
4247:
3886:
3813:
3677:
3578:
288:. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an
81:
1698:
obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a
2672:
1339:
639:
302:
figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a
2728:
is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
1138:
713:
1405:) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel
192:
surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to
1663:
Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.
3942:
4103:
2855:
4242:
2648:
1682:
501:
1329:, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).
3828:
605:{\displaystyle {\begin{aligned}e&=\cos \alpha ,\\a&={\frac {r}{\sin \alpha }}.\end{aligned}}}
20:
4139:
681:
455:
1319:. Equivalently, for a given surface area, the right circular cylinder with the largest volume has
247:
in a plane not parallel to the given line. Any line in this family of parallel lines is called an
4212:
4143:
2772:
2530:
1706:
911:
388:
202:
2848:
1677:
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases
125:
3667:
2135:{\displaystyle A\left(x+{\frac {D}{2A}}\right)^{2}+B\left(y+{\frac {E}{2B}}\right)^{2}=\rho ,}
2031:
does not appear and the general equation of this type of degenerate quadric can be written as
1135:
does not include either top or bottom elements, and therefore has surface area (lateral area)
1074:
of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:
4079:
4021:
2488:
2275:
1406:
299:
236:
177:
4201:
4176:
3110:
2716:
2382:
2354:
This equation of an elliptic cylinder is a generalization of the equation of the ordinary,
2249:
319:
71:
901:{\displaystyle V=\int _{0}^{h}A(x)dx=\int _{0}^{h}\pi abdx=\pi ab\int _{0}^{h}dx=\pi abh.}
500:
In the case of a right circular cylinder with a cylindric section that is an ellipse, the
8:
4171:
4165:
2738:
2721:
2243:
1541:{\displaystyle V=\pi \left(R^{2}-r^{2}\right)h=2\pi \left({\frac {R+r}{2}}\right)h(R-r).}
1336:, of a circular cylinder, which need not be a right cylinder, is more generally given by
415:
219:
158:
4066:
2926:
2835:
1808:. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
315:
185:
4252:
4237:
4045:
3913:
3882:
3875:
3860:
3832:
3809:
3758:
3673:
3632:
3574:
2939:
2931:
2750:
64:
2373:
4027:
2916:
2906:
2896:
2884:
2842:
2801:
2768:
2757:
2478:{\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=-1,}
2024:
1309:
For a given volume, the right circular cylinder with the smallest surface area has
451:
162:
86:
2626:{\displaystyle \left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1.}
2347:{\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.}
4129:
3552:
3526:
3156:
2921:
2911:
2901:
2817:
2784:
2746:
2742:
2725:
1862:
510:
460:. The cylindric section by a plane that contains two elements of a cylinder is a
279:
4091:
3521:
275:
240:
146:
60:
3916:
328:. In some elementary treatments, a cylinder always means a circular cylinder.
4231:
4192:
4148:
4134:
4032:
3072:
3058:
3044:
1831:
478:
461:
338:
3088:
3079:
3065:
4039:
3972:
1703:
1071:
914:, the volume of a right circular cylinder can be calculated by integration
404:
345:
173:
95:
3037:
3051:
2008:
1673:
259:, a cylindrical surface is that surface traced out by a line, called the
244:
214:
The definitions and results in this section are taken from the 1913 text
150:
687:
1695:
1656:{\displaystyle A=2\pi \left(R+r\right)h+2\pi \left(R^{2}-r^{2}\right).}
450:
A cylindric section is the intersection of a cylinder's surface with a
363:
252:
1409:
bases perpendicular to the cylinders' common axis, as in the diagram.
425:
has a height much greater than its diameter, whereas a short and wide
4207:
4096:
3921:
3663:
3637:
3013:
2825:
1868:
1846:
1390:
465:
184:. The shift in the basic meaning—solid versus surface (as in a solid
138:
3020:
1106:
The area of the top and bottom bases is the same, and is called the
677:
This formula holds whether or not the cylinder is a right cylinder.
484:
442:
360:
of the cylinder and it passes through the centers of the two bases.
2971:
2964:
2957:
2877:
2797:
1842:
1710:
1303:
411:
of a right circular cylinder have been known from early antiquity.
181:
3006:
2992:
2978:
1382:
530:
between the secant plane and cylinder axis, in the following way:
3102:
2217:{\displaystyle \rho =-H+{\frac {D^{2}}{4A}}+{\frac {E^{2}}{4B}}.}
1838:
494:
2950:
2274:, then the equation of an elliptic cylinder may be rewritten in
1285:{\displaystyle A=L+2B=2\pi rh+2\pi r^{2}=2\pi r(h+r)=\pi d(r+h)}
3992:
3753:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999),
3603:
3601:
2821:
1699:
633:
620:
408:
189:
166:
107:
2816:
From a polyhedral viewpoint, a cylinder can also be seen as a
518:
of the cylindric section depend on the radius of the cylinder
3998:
154:
3686:
3644:
3613:
3598:
1567:
The surface area, including the top and bottom, is given by
266:
2891:
2800:. The connection is very strong and many older texts treat
2775:
building, Copenhagen, is an example of a truncated cylinder
2000:{\displaystyle f(x,y,z)=Ax^{2}+By^{2}+Cz^{2}+Dx+Ey+Gz+H=0,}
35:
3529:, the intersection of two or three perpendicular cylinders
2999:
2985:
2399:
has a different sign than the coefficients, we obtain the
2372:, but that name is ambiguous, as it can also refer to the
3710:
3698:
1702:
by exploiting the relationship between a sphere and its
454:. They are, in general, curves and are special types of
414:
A right circular cylinder can also be thought of as the
239:
consisting of all the points on all the lines which are
3973:
Compact topological surfaces and their immersions in 3D
3775:
3773:
3586:
1717:
that of the circumscribed cylinder and a surface area
2675:
2563:
2533:
2491:
2409:
2385:
2284:
2252:
2148:
2037:
1882:
1834:
spanned by a one-parameter family of parallel lines.
1573:
1436:
1342:
1178:
1141:
920:
774:
642:
536:
3881:(Alternate ed.), Prindle, Weber & Schmidt,
3785:
3770:
3734:
3722:
464:. Such a cylindric section of a right cylinder is a
691:A solid elliptic right cylinder with the semi-axes
3874:
2706:
2625:
2545:
2503:
2477:
2391:
2346:
2258:
2216:
2134:
1999:
1655:
1540:
1391:Right circular hollow cylinder (cylindrical shell)
1363:
1284:
1162:
1045:
900:
667:
604:
145: 'roller, tumbler') has traditionally been a
3911:
3752:
298:of the cylinder. The two bases of a cylinder are
4229:
3840:(Rev. ed.), Allyn and Bacon, pp. 79–81
1816:In some areas of geometry and topology the term
3895:Wentworth, George; Smith, David Eugene (1913),
1548:Thus, the volume of a cylindrical shell equals
768:the area of each elliptic cross-section, thus:
488:Cylindric sections of a right circular cylinder
348:about a fixed line that it is parallel to is a
255:point of view, given a plane curve, called the
243:to a given line and which pass through a fixed
3894:
3692:
3650:
3619:
3607:
223:
3958:
3834:Solid Geometry with Problems and Applications
3827:
2856:
1837:A cylinder having a right section that is an
1666:
429:has a diameter much greater than its height.
2760:, which may include the cylindrical conics.
2242:. Further simplification can be obtained by
382:
180:in various modern branches of geometry and
3965:
3951:
3757:, Cambridge University Press, p. 34,
3662:
2863:
2849:
34:
3872:
3716:
3704:
3592:
1032:
1021:
1001:
994:
987:
986:
680:This formula may be established by using
619:If the base of a circular cylinder has a
278:bounded by a cylindrical surface and two
2767:
2756:This concept is useful when considering
2715:
2368:). Elliptic cylinders are also known as
1867:
1672:
1381:
686:
483:
441:
362:
270:A right and an oblique circular cylinder
265:
2669:with equations that can be written as:
2557:, whose equations may be rewritten as:
1811:
1116:. The area of the side is known as the
4230:
3854:
3791:
3779:
3740:
3728:
3573:, W. H. Freeman and Co., p. 607,
3568:
3555:, Henry George Liddell, Robert Scott,
2783:can be seen as the limiting case of a
2732:
2514:
2266:has the same sign as the coefficients
1687:In the treatise by this name, written
367:A right circular cylinder with radius
3946:
3912:
3803:
2635:
1861:, respectively. These are degenerate
1774:. The surface area of this sphere is
437:
172:A cylinder may also be defined as an
2724:, a cylinder is simply a cone whose
2485:which have no real points on them. (
2226:
344:The cylinder obtained by rotating a
40:A circular right cylinder of height
741:is the area of the base ellipse (=
337:(or altitude) of a cylinder is the
251:of the cylindrical surface. From a
13:
14:
4264:
3905:
1820:refers to what has been called a
3855:Albert, Abraham Adrian (2016) ,
3504:
3499:
3494:
3489:
3484:
3472:
3467:
3462:
3457:
3452:
3443:
3438:
3433:
3428:
3423:
3414:
3409:
3404:
3399:
3394:
3385:
3380:
3375:
3370:
3365:
3356:
3351:
3346:
3341:
3336:
3327:
3322:
3317:
3312:
3307:
3298:
3293:
3288:
3283:
3278:
3269:
3264:
3259:
3254:
3249:
3240:
3235:
3230:
3225:
3220:
3211:
3206:
3201:
3196:
3191:
3182:
3177:
3172:
3167:
3162:
3101:
3087:
3078:
3071:
3064:
3057:
3050:
3043:
3036:
3019:
3012:
3005:
2998:
2991:
2984:
2977:
2970:
2963:
2956:
2949:
1375:is the length of an element and
703:for the base ellipse and height
3877:Calculus with Analytic Geometry
3821:
3806:Geometry a Comprehensive Course
3797:
3746:
1306:of the circular top or bottom.
1055:
3666:; Terrell, Maria Shea (2013),
3656:
3625:
3562:
3539:
2246:and scalar multiplication. If
1904:
1886:
1532:
1520:
1397:right circular hollow cylinder
1279:
1267:
1255:
1243:
805:
799:
1:
3847:
1688:
1087:the area of the bottom base:
509:of the cylindric section and
432:
318:(regions whose boundary is a
2707:{\displaystyle x^{2}+2ay=0.}
2511:gives a single real point.)
2401:imaginary elliptic cylinders
2007:with the coefficients being
1364:{\displaystyle L=e\times p,}
668:{\displaystyle V=\pi r^{2}h}
628:and the cylinder has height
341:distance between its bases.
308:, otherwise it is called an
7:
3873:Swokowski, Earl W. (1983),
3515:
2546:{\displaystyle \rho \neq 0}
2238:this is the equation of an
322:) the cylinder is called a
149:, one of the most basic of
10:
4269:
3931:Surface area of a cylinder
3693:Wentworth & Smith 1913
3669:Calculus With Applications
3651:Wentworth & Smith 1913
3620:Wentworth & Smith 1913
3608:Wentworth & Smith 1913
3569:Jacobs, Harold R. (1974),
2833:
2649:without loss of generality
1713:. The sphere has a volume
1683:On the Sphere and Cylinder
1680:
1668:On the Sphere and Cylinder
1430:. The volume is given by
1078:the area of the top base:
493:cylindrical surface in an
386:
224:Wentworth & Smith 1913
131:
18:
4185:
4157:
4122:
4113:
4059:
4014:
3985:
3978:
3672:, Springer, p. 178,
2763:
2741:, a cylinder is simply a
2527:have different signs and
1163:{\displaystyle L=2\pi rh}
614:
106:
94:
80:
70:
56:
33:
28:
21:Cylinder (disambiguation)
4248:Euclidean solid geometry
3897:Plane and Solid Geometry
3533:
383:Right circular cylinders
222:and David Eugene Smith (
216:Plane and Solid Geometry
209:
4104:Sphere with three holes
3857:Solid Analytic Geometry
3831:; Lennes, N.J. (1919),
3557:A Greek-English Lexicon
3033:Spherical tiling image
2781:solid circular cylinder
2773:Tycho Brahe Planetarium
2504:{\displaystyle \rho =0}
1709:of the same height and
1707:right circular cylinder
912:cylindrical coordinates
403:. The formulae for the
389:Right circular cylinder
290:element of the cylinder
203:right circular cylinder
147:three-dimensional solid
16:Three-dimensional solid
3808:, Dover, p. 398,
2776:
2729:
2708:
2627:
2547:
2505:
2479:
2393:
2348:
2260:
2218:
2136:
2001:
1873:
1678:
1657:
1558:average radius ×
1542:
1424:, and external radius
1387:
1365:
1286:
1164:
1096:the area of the side:
1066:and altitude (height)
1047:
902:
709:
669:
606:
489:
447:
379:
351:cylinder of revolution
271:
4022:Real projective plane
4007:Pretzel (genus 3) ...
2824:as an infinite-sided
2771:
2749:(vertex) lies on the
2719:
2709:
2628:
2548:
2506:
2480:
2394:
2392:{\displaystyle \rho }
2349:
2276:Cartesian coordinates
2261:
2259:{\displaystyle \rho }
2219:
2137:
2002:
1871:
1827:generalized cylinders
1676:
1658:
1543:
1385:
1366:
1287:
1165:
1048:
903:
690:
682:Cavalieri's principle
670:
607:
487:
445:
366:
269:
161:, it is considered a
4177:Euler characteristic
3937:Volume of a cylinder
3804:Pedoe, Dan (1988) ,
2673:
2561:
2555:hyperbolic cylinders
2531:
2489:
2407:
2383:
2282:
2250:
2146:
2035:
1880:
1812:Cylindrical surfaces
1571:
1434:
1340:
1176:
1139:
918:
772:
640:
534:
282:is called a (solid)
198:cylindrical surfaces
19:For other uses, see
3633:"Cylindric section"
3098:Plane tiling image
2739:projective geometry
2733:Projective geometry
2722:projective geometry
2667:parabolic cylinders
2515:Hyperbolic cylinder
2244:translation of axes
1859:hyperbolic cylinder
1822:cylindrical surface
982:
967:
949:
873:
831:
795:
416:solid of revolution
314:. If the bases are
232:cylindrical surface
220:George A. Wentworth
159:elementary geometry
4004:Number 8 (genus 2)
3914:Weisstein, Eric W.
3551:2013-07-30 at the
2927:Hendecagonal prism
2811:truncated cylinder
2777:
2730:
2704:
2636:Parabolic cylinder
2623:
2543:
2501:
2475:
2389:
2344:
2256:
2214:
2132:
2027:that the variable
1997:
1874:
1872:Parabolic cylinder
1855:parabolic cylinder
1679:
1653:
1564: thickness.
1538:
1418:, internal radius
1412:Let the height be
1388:
1361:
1332:The lateral area,
1282:
1160:
1043:
1041:
968:
950:
935:
898:
859:
817:
781:
719:, semi-minor axis
710:
665:
602:
600:
490:
448:
438:Cylindric sections
380:
272:
4243:Elementary shapes
4225:
4224:
4221:
4220:
4055:
4054:
3866:978-0-486-81026-3
3764:978-0-521-59787-6
3513:
3512:
2946:Polyhedron image
2940:Apeirogonal prism
2932:Dodecagonal prism
2758:degenerate conics
2751:plane at infinity
2605:
2577:
2451:
2423:
2356:circular cylinder
2326:
2298:
2240:elliptic cylinder
2227:Elliptic cylinder
2209:
2184:
2110:
2066:
1851:elliptic cylinder
1511:
1402:cylindrical shell
593:
446:Cylindric section
325:circular cylinder
118:
117:
65:Algebraic surface
4260:
4140:Triangulatedness
4120:
4119:
3983:
3982:
3979:Without boundary
3967:
3960:
3953:
3944:
3943:
3927:
3926:
3900:
3891:
3880:
3869:
3842:
3841:
3839:
3825:
3819:
3818:
3801:
3795:
3789:
3783:
3777:
3768:
3767:
3750:
3744:
3738:
3732:
3726:
3720:
3714:
3708:
3702:
3696:
3690:
3684:
3682:
3660:
3654:
3648:
3642:
3641:
3629:
3623:
3617:
3611:
3605:
3596:
3590:
3584:
3583:
3566:
3560:
3543:
3509:
3508:
3507:
3503:
3502:
3498:
3497:
3493:
3492:
3488:
3487:
3477:
3476:
3475:
3471:
3470:
3466:
3465:
3461:
3460:
3456:
3455:
3448:
3447:
3446:
3442:
3441:
3437:
3436:
3432:
3431:
3427:
3426:
3419:
3418:
3417:
3413:
3412:
3408:
3407:
3403:
3402:
3398:
3397:
3390:
3389:
3388:
3384:
3383:
3379:
3378:
3374:
3373:
3369:
3368:
3361:
3360:
3359:
3355:
3354:
3350:
3349:
3345:
3344:
3340:
3339:
3332:
3331:
3330:
3326:
3325:
3321:
3320:
3316:
3315:
3311:
3310:
3303:
3302:
3301:
3297:
3296:
3292:
3291:
3287:
3286:
3282:
3281:
3274:
3273:
3272:
3268:
3267:
3263:
3262:
3258:
3257:
3253:
3252:
3245:
3244:
3243:
3239:
3238:
3234:
3233:
3229:
3228:
3224:
3223:
3216:
3215:
3214:
3210:
3209:
3205:
3204:
3200:
3199:
3195:
3194:
3187:
3186:
3185:
3181:
3180:
3176:
3175:
3171:
3170:
3166:
3165:
3105:
3091:
3082:
3075:
3068:
3061:
3054:
3047:
3040:
3023:
3016:
3009:
3002:
2995:
2988:
2981:
2974:
2967:
2960:
2953:
2917:Enneagonal prism
2907:Heptagonal prism
2897:Pentagonal prism
2885:Triangular prism
2865:
2858:
2851:
2831:
2830:
2795:
2787:
2713:
2711:
2710:
2705:
2685:
2684:
2664:
2657:
2646:
2632:
2630:
2629:
2624:
2616:
2615:
2610:
2606:
2598:
2588:
2587:
2582:
2578:
2570:
2553:, we obtain the
2552:
2550:
2549:
2544:
2526:
2522:
2510:
2508:
2507:
2502:
2484:
2482:
2481:
2476:
2462:
2461:
2456:
2452:
2444:
2434:
2433:
2428:
2424:
2416:
2398:
2396:
2395:
2390:
2367:
2353:
2351:
2350:
2345:
2337:
2336:
2331:
2327:
2319:
2309:
2308:
2303:
2299:
2291:
2273:
2269:
2265:
2263:
2262:
2257:
2237:
2223:
2221:
2220:
2215:
2210:
2208:
2200:
2199:
2190:
2185:
2183:
2175:
2174:
2165:
2141:
2139:
2138:
2133:
2122:
2121:
2116:
2112:
2111:
2109:
2098:
2078:
2077:
2072:
2068:
2067:
2065:
2054:
2030:
2025:rotation of axes
2022:
2018:
2014:
2006:
2004:
2003:
1998:
1954:
1953:
1938:
1937:
1922:
1921:
1863:quadric surfaces
1807:
1802:
1798:
1796:
1795:
1792:
1789:
1779:
1773:
1768:
1764:
1762:
1761:
1758:
1755:
1745:
1742:
1740:
1739:
1736:
1733:
1725:
1720:
1716:
1693:
1690:
1662:
1660:
1659:
1654:
1649:
1645:
1644:
1643:
1631:
1630:
1604:
1600:
1563:
1562:altitude ×
1559:
1555:
1553:
1547:
1545:
1544:
1539:
1516:
1512:
1507:
1496:
1478:
1474:
1473:
1472:
1460:
1459:
1429:
1423:
1417:
1378:
1374:
1370:
1368:
1367:
1362:
1335:
1328:
1318:
1301:
1291:
1289:
1288:
1283:
1230:
1229:
1169:
1167:
1166:
1161:
1127:
1115:
1102:
1093:
1084:
1069:
1065:
1052:
1050:
1049:
1044:
1042:
1031:
1030:
1011:
981:
976:
966:
958:
948:
943:
907:
905:
904:
899:
872:
867:
830:
825:
794:
789:
767:
753:
749:
745:
740:
736:
726:
722:
718:
708:
702:
696:
674:
672:
671:
666:
661:
660:
631:
627:
611:
609:
608:
603:
601:
594:
592:
578:
529:
523:
517:
508:
421:A tall and thin
378:
372:
311:oblique cylinder
142:
135:
114:
102:
89:
38:
26:
25:
4268:
4267:
4263:
4262:
4261:
4259:
4258:
4257:
4228:
4227:
4226:
4217:
4181:
4158:Characteristics
4153:
4115:
4109:
4051:
4010:
3974:
3971:
3908:
3903:
3889:
3867:
3850:
3845:
3837:
3826:
3822:
3816:
3802:
3798:
3790:
3786:
3778:
3771:
3765:
3751:
3747:
3739:
3735:
3727:
3723:
3715:
3711:
3703:
3699:
3691:
3687:
3680:
3661:
3657:
3649:
3645:
3631:
3630:
3626:
3618:
3614:
3606:
3599:
3591:
3587:
3581:
3567:
3563:
3553:Wayback Machine
3544:
3540:
3536:
3527:Steinmetz solid
3518:
3505:
3500:
3495:
3490:
3485:
3483:
3473:
3468:
3463:
3458:
3453:
3451:
3444:
3439:
3434:
3429:
3424:
3422:
3415:
3410:
3405:
3400:
3395:
3393:
3386:
3381:
3376:
3371:
3366:
3364:
3357:
3352:
3347:
3342:
3337:
3335:
3328:
3323:
3318:
3313:
3308:
3306:
3299:
3294:
3289:
3284:
3279:
3277:
3270:
3265:
3260:
3255:
3250:
3248:
3241:
3236:
3231:
3226:
3221:
3219:
3212:
3207:
3202:
3197:
3192:
3190:
3183:
3178:
3173:
3168:
3163:
3161:
3157:Coxeter diagram
2922:Decagonal prism
2912:Octagonal prism
2902:Hexagonal prism
2890:
2883:
2869:
2807:truncated prism
2791:
2785:
2766:
2735:
2680:
2676:
2674:
2671:
2670:
2659:
2652:
2641:
2638:
2611:
2597:
2593:
2592:
2583:
2569:
2565:
2564:
2562:
2559:
2558:
2532:
2529:
2528:
2524:
2520:
2517:
2490:
2487:
2486:
2457:
2443:
2439:
2438:
2429:
2415:
2411:
2410:
2408:
2405:
2404:
2384:
2381:
2380:
2359:
2332:
2318:
2314:
2313:
2304:
2290:
2286:
2285:
2283:
2280:
2279:
2271:
2267:
2251:
2248:
2247:
2232:
2229:
2201:
2195:
2191:
2189:
2176:
2170:
2166:
2164:
2147:
2144:
2143:
2117:
2102:
2097:
2090:
2086:
2085:
2073:
2058:
2053:
2046:
2042:
2041:
2036:
2033:
2032:
2028:
2020:
2016:
2012:
2011:and not all of
1949:
1945:
1933:
1929:
1917:
1913:
1881:
1878:
1877:
1828:
1814:
1800:
1793:
1790:
1787:
1786:
1784:
1777:
1775:
1766:
1759:
1756:
1753:
1752:
1750:
1743:
1737:
1734:
1731:
1730:
1728:
1727:
1723:
1718:
1714:
1691:
1685:
1671:
1639:
1635:
1626:
1622:
1621:
1617:
1590:
1586:
1572:
1569:
1568:
1561:
1557:
1551:
1549:
1497:
1495:
1491:
1468:
1464:
1455:
1451:
1450:
1446:
1435:
1432:
1431:
1425:
1419:
1413:
1403:
1393:
1386:Hollow cylinder
1376:
1372:
1341:
1338:
1337:
1333:
1320:
1310:
1293:
1225:
1221:
1177:
1174:
1173:
1140:
1137:
1136:
1123:
1120:
1111:
1097:
1088:
1079:
1067:
1061:
1058:
1040:
1039:
1026:
1022:
1009:
1008:
977:
972:
959:
954:
944:
939:
928:
921:
919:
916:
915:
868:
863:
826:
821:
790:
785:
773:
770:
769:
755:
751:
743:
742:
738:
728:
724:
720:
716:
714:semi-major axis
704:
698:
692:
656:
652:
641:
638:
637:
629:
623:
617:
599:
598:
582:
577:
570:
564:
563:
544:
537:
535:
532:
531:
525:
519:
513:
511:semi-major axis
504:
475:
440:
435:
423:needle cylinder
401:
391:
385:
374:
368:
358:
352:
335:
326:
312:
306:
296:
286:
280:parallel planes
233:
212:
194:solid cylinders
112:
100:
87:
63:
52:
24:
17:
12:
11:
5:
4266:
4256:
4255:
4250:
4245:
4240:
4223:
4222:
4219:
4218:
4216:
4215:
4210:
4204:
4198:
4195:
4189:
4187:
4183:
4182:
4180:
4179:
4174:
4169:
4161:
4159:
4155:
4154:
4152:
4151:
4146:
4137:
4132:
4126:
4124:
4117:
4111:
4110:
4108:
4107:
4101:
4100:
4099:
4089:
4088:
4087:
4082:
4074:
4073:
4072:
4063:
4061:
4057:
4056:
4053:
4052:
4050:
4049:
4046:Dyck's surface
4043:
4037:
4036:
4035:
4030:
4018:
4016:
4015:Non-orientable
4012:
4011:
4009:
4008:
4005:
4002:
3996:
3989:
3987:
3980:
3976:
3975:
3970:
3969:
3962:
3955:
3947:
3941:
3940:
3934:
3928:
3907:
3906:External links
3904:
3902:
3901:
3899:, Ginn and Co.
3892:
3887:
3870:
3865:
3851:
3849:
3846:
3844:
3843:
3820:
3814:
3796:
3784:
3769:
3763:
3745:
3733:
3721:
3719:, p. 291.
3717:Swokowski 1983
3709:
3707:, p. 292.
3705:Swokowski 1983
3697:
3695:, p. 358.
3685:
3678:
3655:
3653:, p. 359.
3643:
3624:
3622:, p. 357.
3612:
3610:, p. 354.
3597:
3595:, p. 283.
3593:Swokowski 1983
3585:
3579:
3561:
3537:
3535:
3532:
3531:
3530:
3524:
3522:List of shapes
3517:
3514:
3511:
3510:
3481:
3478:
3449:
3420:
3391:
3362:
3333:
3304:
3275:
3246:
3217:
3188:
3159:
3153:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3111:Vertex config.
3107:
3106:
3099:
3096:
3094:
3092:
3085:
3083:
3076:
3069:
3062:
3055:
3048:
3041:
3034:
3030:
3029:
3027:
3024:
3017:
3010:
3003:
2996:
2989:
2982:
2975:
2968:
2961:
2954:
2947:
2943:
2942:
2937:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2887:
2880:
2875:
2871:
2870:
2868:
2867:
2860:
2853:
2845:
2765:
2762:
2734:
2731:
2703:
2700:
2697:
2694:
2691:
2688:
2683:
2679:
2665:to obtain the
2637:
2634:
2622:
2619:
2614:
2609:
2604:
2601:
2596:
2591:
2586:
2581:
2576:
2573:
2568:
2542:
2539:
2536:
2516:
2513:
2500:
2497:
2494:
2474:
2471:
2468:
2465:
2460:
2455:
2450:
2447:
2442:
2437:
2432:
2427:
2422:
2419:
2414:
2388:
2374:Plücker conoid
2343:
2340:
2335:
2330:
2325:
2322:
2317:
2312:
2307:
2302:
2297:
2294:
2289:
2255:
2228:
2225:
2213:
2207:
2204:
2198:
2194:
2188:
2182:
2179:
2173:
2169:
2163:
2160:
2157:
2154:
2151:
2131:
2128:
2125:
2120:
2115:
2108:
2105:
2101:
2096:
2093:
2089:
2084:
2081:
2076:
2071:
2064:
2061:
2057:
2052:
2049:
2045:
2040:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1952:
1948:
1944:
1941:
1936:
1932:
1928:
1925:
1920:
1916:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1826:
1813:
1810:
1692: 225 BCE
1681:Main article:
1670:
1665:
1652:
1648:
1642:
1638:
1634:
1629:
1625:
1620:
1616:
1613:
1610:
1607:
1603:
1599:
1596:
1593:
1589:
1585:
1582:
1579:
1576:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1515:
1510:
1506:
1503:
1500:
1494:
1490:
1487:
1484:
1481:
1477:
1471:
1467:
1463:
1458:
1454:
1449:
1445:
1442:
1439:
1401:
1392:
1389:
1360:
1357:
1354:
1351:
1348:
1345:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1228:
1224:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1159:
1156:
1153:
1150:
1147:
1144:
1118:
1104:
1103:
1094:
1085:
1060:Having radius
1057:
1054:
1038:
1035:
1029:
1025:
1020:
1017:
1014:
1012:
1010:
1007:
1004:
1000:
997:
993:
990:
985:
980:
975:
971:
965:
962:
957:
953:
947:
942:
938:
934:
931:
929:
927:
924:
923:
897:
894:
891:
888:
885:
882:
879:
876:
871:
866:
862:
858:
855:
852:
849:
846:
843:
840:
837:
834:
829:
824:
820:
816:
813:
810:
807:
804:
801:
798:
793:
788:
784:
780:
777:
664:
659:
655:
651:
648:
645:
616:
613:
597:
591:
588:
585:
581:
576:
573:
571:
569:
566:
565:
562:
559:
556:
553:
550:
547:
545:
543:
540:
539:
524:and the angle
473:
457:plane sections
439:
436:
434:
431:
399:
393:The bare term
387:Main article:
384:
381:
356:
350:
333:
324:
310:
305:right cylinder
304:
294:
284:
231:
211:
208:
116:
115:
110:
104:
103:
98:
92:
91:
84:
82:Symmetry group
78:
77:
74:
68:
67:
61:Smooth surface
58:
54:
53:
39:
31:
30:
15:
9:
6:
4:
3:
2:
4265:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4235:
4233:
4214:
4211:
4209:
4205:
4203:
4199:
4197:Making a hole
4196:
4194:
4193:Connected sum
4191:
4190:
4188:
4184:
4178:
4175:
4173:
4170:
4167:
4163:
4162:
4160:
4156:
4150:
4149:Orientability
4147:
4145:
4141:
4138:
4136:
4133:
4131:
4130:Connectedness
4128:
4127:
4125:
4121:
4118:
4112:
4105:
4102:
4098:
4095:
4094:
4093:
4090:
4086:
4083:
4081:
4078:
4077:
4075:
4070:
4069:
4068:
4065:
4064:
4062:
4060:With boundary
4058:
4048:(genus 3) ...
4047:
4044:
4041:
4038:
4034:
4033:Roman surface
4031:
4029:
4028:Boy's surface
4025:
4024:
4023:
4020:
4019:
4017:
4013:
4006:
4003:
4000:
3997:
3994:
3991:
3990:
3988:
3984:
3981:
3977:
3968:
3963:
3961:
3956:
3954:
3949:
3948:
3945:
3938:
3935:
3932:
3929:
3924:
3923:
3918:
3915:
3910:
3909:
3898:
3893:
3890:
3888:0-87150-341-7
3884:
3879:
3878:
3871:
3868:
3862:
3858:
3853:
3852:
3836:
3835:
3830:
3829:Slaught, H.E.
3824:
3817:
3815:0-486-65812-0
3811:
3807:
3800:
3794:, p. 75.
3793:
3788:
3782:, p. 74.
3781:
3776:
3774:
3766:
3760:
3756:
3749:
3743:, p. 49.
3742:
3737:
3731:, p. 43.
3730:
3725:
3718:
3713:
3706:
3701:
3694:
3689:
3681:
3679:9781461479468
3675:
3671:
3670:
3665:
3664:Lax, Peter D.
3659:
3652:
3647:
3640:
3639:
3634:
3628:
3621:
3616:
3609:
3604:
3602:
3594:
3589:
3582:
3580:0-7167-0456-0
3576:
3572:
3565:
3558:
3554:
3550:
3547:
3542:
3538:
3528:
3525:
3523:
3520:
3519:
3482:
3479:
3450:
3421:
3392:
3363:
3334:
3305:
3276:
3247:
3218:
3189:
3160:
3158:
3155:
3154:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3112:
3109:
3108:
3104:
3100:
3097:
3095:
3093:
3090:
3086:
3084:
3081:
3077:
3074:
3070:
3067:
3063:
3060:
3056:
3053:
3049:
3046:
3042:
3039:
3035:
3032:
3031:
3028:
3025:
3022:
3018:
3015:
3011:
3008:
3004:
3001:
2997:
2994:
2990:
2987:
2983:
2980:
2976:
2973:
2969:
2966:
2962:
2959:
2955:
2952:
2948:
2945:
2944:
2941:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2888:
2886:
2881:
2879:
2878:Digonal prism
2876:
2873:
2872:
2866:
2861:
2859:
2854:
2852:
2847:
2846:
2844:
2840:
2837:
2832:
2829:
2827:
2823:
2819:
2814:
2812:
2808:
2803:
2799:
2794:
2789:
2782:
2774:
2770:
2761:
2759:
2754:
2752:
2748:
2744:
2740:
2727:
2723:
2718:
2714:
2701:
2698:
2695:
2692:
2689:
2686:
2681:
2677:
2668:
2662:
2655:
2650:
2644:
2633:
2620:
2617:
2612:
2607:
2602:
2599:
2594:
2589:
2584:
2579:
2574:
2571:
2566:
2556:
2540:
2537:
2534:
2512:
2498:
2495:
2492:
2472:
2469:
2466:
2463:
2458:
2453:
2448:
2445:
2440:
2435:
2430:
2425:
2420:
2417:
2412:
2402:
2386:
2377:
2375:
2371:
2366:
2362:
2357:
2341:
2338:
2333:
2328:
2323:
2320:
2315:
2310:
2305:
2300:
2295:
2292:
2287:
2277:
2253:
2245:
2241:
2235:
2224:
2211:
2205:
2202:
2196:
2192:
2186:
2180:
2177:
2171:
2167:
2161:
2158:
2155:
2152:
2149:
2129:
2126:
2123:
2118:
2113:
2106:
2103:
2099:
2094:
2091:
2087:
2082:
2079:
2074:
2069:
2062:
2059:
2055:
2050:
2047:
2043:
2038:
2026:
2010:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1950:
1946:
1942:
1939:
1934:
1930:
1926:
1923:
1918:
1914:
1910:
1907:
1901:
1898:
1895:
1892:
1889:
1883:
1870:
1866:
1864:
1860:
1856:
1852:
1849:is called an
1848:
1844:
1840:
1835:
1833:
1832:ruled surface
1829:
1823:
1819:
1809:
1805:
1782:
1771:
1748:
1712:
1708:
1705:
1704:circumscribed
1701:
1697:
1684:
1675:
1669:
1664:
1650:
1646:
1640:
1636:
1632:
1627:
1623:
1618:
1614:
1611:
1608:
1605:
1601:
1597:
1594:
1591:
1587:
1583:
1580:
1577:
1574:
1565:
1535:
1529:
1526:
1523:
1517:
1513:
1508:
1504:
1501:
1498:
1492:
1488:
1485:
1482:
1479:
1475:
1469:
1465:
1461:
1456:
1452:
1447:
1443:
1440:
1437:
1428:
1422:
1416:
1410:
1408:
1404:
1398:
1384:
1380:
1358:
1355:
1352:
1349:
1346:
1343:
1330:
1327:
1323:
1317:
1313:
1307:
1305:
1300:
1296:
1276:
1273:
1270:
1264:
1261:
1258:
1252:
1249:
1246:
1240:
1237:
1234:
1231:
1226:
1222:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1170:
1157:
1154:
1151:
1148:
1145:
1142:
1134:
1133:open cylinder
1129:
1126:
1121:
1114:
1109:
1101:
1095:
1092:
1086:
1083:
1077:
1076:
1075:
1073:
1064:
1053:
1036:
1033:
1027:
1023:
1018:
1015:
1013:
1005:
1002:
998:
995:
991:
988:
983:
978:
973:
969:
963:
960:
955:
951:
945:
940:
936:
932:
930:
925:
913:
908:
895:
892:
889:
886:
883:
880:
877:
874:
869:
864:
860:
856:
853:
850:
847:
844:
841:
838:
835:
832:
827:
822:
818:
814:
811:
808:
802:
796:
791:
786:
782:
778:
775:
766:
762:
758:
748:
735:
731:
727:has a volume
715:
707:
701:
695:
689:
685:
683:
678:
675:
662:
657:
653:
649:
646:
643:
635:
626:
622:
612:
595:
589:
586:
583:
579:
574:
572:
567:
560:
557:
554:
551:
548:
546:
541:
528:
522:
516:
512:
507:
503:
498:
496:
486:
482:
480:
479:conic section
476:
474:right section
469:
467:
463:
462:parallelogram
459:
458:
453:
444:
430:
428:
427:disk cylinder
424:
419:
417:
412:
410:
406:
402:
400:open cylinder
396:
390:
377:
371:
365:
361:
359:
353:
347:
342:
340:
339:perpendicular
336:
329:
327:
321:
317:
313:
307:
301:
297:
291:
287:
281:
277:
268:
264:
262:
258:
254:
250:
246:
242:
238:
234:
227:
225:
221:
217:
207:
205:
204:
199:
195:
191:
187:
183:
179:
175:
170:
169:as its base.
168:
164:
160:
156:
152:
148:
144:
141:
134:
130:
127:
126:Ancient Greek
123:
111:
109:
105:
99:
97:
93:
90:
85:
83:
79:
75:
73:
69:
66:
62:
59:
55:
51:
47:
44:and diameter
43:
37:
32:
27:
22:
4092:Möbius strip
4084:
4040:Klein bottle
3939:at MATHguide
3933:at MATHguide
3920:
3896:
3876:
3856:
3833:
3823:
3805:
3799:
3787:
3754:
3748:
3736:
3724:
3712:
3700:
3688:
3668:
3658:
3646:
3636:
3627:
3615:
3588:
3570:
3564:
3559:, on Perseus
3556:
3541:
2892:Square prism
2889:(Tetragonal)
2838:
2815:
2810:
2806:
2792:
2790:prism where
2780:
2778:
2755:
2736:
2666:
2660:
2653:
2642:
2640:Finally, if
2639:
2554:
2518:
2400:
2378:
2369:
2364:
2360:
2355:
2239:
2233:
2230:
2009:real numbers
1875:
1858:
1854:
1850:
1836:
1825:
1821:
1817:
1815:
1803:
1780:
1769:
1746:
1686:
1667:
1566:
1426:
1420:
1414:
1411:
1400:
1396:
1394:
1331:
1325:
1321:
1315:
1311:
1308:
1298:
1294:
1171:
1132:
1130:
1124:
1119:lateral area
1117:
1112:
1107:
1105:
1099:
1090:
1081:
1072:surface area
1062:
1059:
1056:Surface area
909:
764:
760:
756:
746:
733:
729:
711:
705:
699:
693:
679:
676:
636:is given by
624:
618:
526:
520:
514:
505:
502:eccentricity
499:
491:
472:
470:
456:
449:
426:
422:
420:
413:
405:surface area
398:
394:
392:
375:
369:
355:
349:
346:line segment
343:
332:
330:
323:
309:
303:
293:
289:
283:
273:
260:
256:
248:
230:
228:
215:
213:
201:
197:
193:
176:curvilinear
171:
139:
136:
129:
121:
119:
96:Surface area
49:
45:
41:
4135:Compactness
3792:Albert 2016
3780:Albert 2016
3741:Albert 2016
3729:Albert 2016
2874:Prism name
2796:approaches
2370:cylindroids
723:and height
632:, then its
373:and height
245:plane curve
151:curvilinear
72:Euler char.
4232:Categories
4186:Operations
4168:components
4164:Number of
4144:smoothness
4123:Properties
4071:Semisphere
3986:Orientable
3917:"Cylinder"
3848:References
2882:(Trigonal)
2834:Family of
1719:two-thirds
1715:two-thirds
1696:Archimedes
754:-axis and
433:Properties
261:generatrix
253:kinematics
153:geometric
124:(from
101:2πr(r + h)
4213:Immersion
4208:cross-cap
4206:Gluing a
4200:Gluing a
4097:Cross-cap
4042:(genus 2)
4026:genus 1;
4001:(genus 1)
3995:(genus 0)
3922:MathWorld
3859:, Dover,
3638:MathWorld
3546:κύλινδρος
2826:bipyramid
2590:−
2538:≠
2535:ρ
2493:ρ
2467:−
2387:ρ
2254:ρ
2156:−
2150:ρ
2127:ρ
1847:hyperbola
1633:−
1615:π
1584:π
1527:−
1489:π
1462:−
1444:π
1353:×
1262:π
1238:π
1219:π
1204:π
1152:π
1108:base area
1019:π
999:ϕ
970:∫
964:π
952:∫
937:∫
884:π
861:∫
851:π
833:π
819:∫
783:∫
650:π
590:α
587:
558:α
555:
466:rectangle
300:congruent
257:directrix
140:kúlindros
133:κύλινδρος
88:O(2)×O(1)
4253:Surfaces
4238:Quadrics
4166:boundary
4085:Cylinder
3755:Geometry
3571:Geometry
3549:Archived
3516:See also
2798:infinity
2647:assume,
1843:parabola
1818:cylinder
1711:diameter
1554: ×
1304:diameter
737:, where
407:and the
395:cylinder
285:cylinder
241:parallel
182:topology
174:infinite
122:cylinder
29:Cylinder
4116:notions
4114:Related
4080:Annulus
4076:Ribbon
2841:-gonal
2836:uniform
2651:, that
1839:ellipse
1797:
1785:
1763:
1751:
1741:
1729:
1560:
1556:
1407:annular
1302:is the
495:ellipse
249:element
237:surface
188:versus
178:surface
165:with a
4202:handle
3993:Sphere
3885:
3863:
3812:
3761:
3676:
3577:
3151:∞.4.4
3145:12.4.4
3142:11.4.4
3139:10.4.4
2843:prisms
2822:bicone
2802:prisms
2788:-gonal
2764:Prisms
2745:whose
2236:> 0
2142:where
1700:sphere
1371:where
1292:where
1070:, the
910:Using
634:volume
621:radius
615:Volume
409:volume
334:height
320:circle
190:sphere
167:circle
155:shapes
108:Volume
4172:Genus
3999:Torus
3838:(PDF)
3534:Notes
3136:9.4.4
3133:8.4.4
3130:7.4.4
3127:6.4.4
3124:5.4.4
3121:4.4.4
3118:3.4.4
3115:2.4.4
2820:of a
1845:, or
452:plane
316:disks
276:solid
235:is a
210:Types
163:prism
157:. In
128:
4067:Disk
3883:ISBN
3861:ISBN
3810:ISBN
3759:ISBN
3674:ISBN
3575:ISBN
3480:...
3026:...
2936:...
2818:dual
2747:apex
2743:cone
2726:apex
2658:and
2523:and
2278:as:
2270:and
2019:and
1857:and
1399:(or
763:) =
697:and
357:axis
331:The
295:base
196:and
186:ball
57:Type
4142:or
4106:...
3148:...
2737:In
2720:In
2663:= 1
2656:= 0
2645:= 0
2519:If
2379:If
2231:If
1726:is
1324:= 2
1314:= 2
1297:= 2
1131:An
584:sin
552:cos
226:).
218:by
113:πrh
4234::
3919:.
3772:^
3635:,
3600:^
2828:.
2813:.
2779:A
2702:0.
2643:AB
2621:1.
2403::
2376:.
2363:=
2342:1.
2234:AB
2015:,
1865:.
1853:,
1841:,
1799:(6
1783:=
1765:(2
1749:=
1694:,
1689:c.
1395:A
1128:.
1122:,
1110:,
1100:rh
1098:2π
747:ab
734:Ah
732:=
684:.
468:.
274:A
229:A
206:.
120:A
48:=2
3966:e
3959:t
3952:v
3925:.
3683:.
2864:e
2857:t
2850:v
2839:n
2793:n
2786:n
2699:=
2696:y
2693:a
2690:2
2687:+
2682:2
2678:x
2661:A
2654:B
2618:=
2613:2
2608:)
2603:b
2600:y
2595:(
2585:2
2580:)
2575:a
2572:x
2567:(
2541:0
2525:B
2521:A
2499:0
2496:=
2473:,
2470:1
2464:=
2459:2
2454:)
2449:b
2446:y
2441:(
2436:+
2431:2
2426:)
2421:a
2418:x
2413:(
2365:b
2361:a
2358:(
2339:=
2334:2
2329:)
2324:b
2321:y
2316:(
2311:+
2306:2
2301:)
2296:a
2293:x
2288:(
2272:B
2268:A
2212:.
2206:B
2203:4
2197:2
2193:E
2187:+
2181:A
2178:4
2172:2
2168:D
2162:+
2159:H
2153:=
2130:,
2124:=
2119:2
2114:)
2107:B
2104:2
2100:E
2095:+
2092:y
2088:(
2083:B
2080:+
2075:2
2070:)
2063:A
2060:2
2056:D
2051:+
2048:x
2044:(
2039:A
2029:z
2021:C
2017:B
2013:A
1995:,
1992:0
1989:=
1986:H
1983:+
1980:z
1977:G
1974:+
1971:y
1968:E
1965:+
1962:x
1959:D
1956:+
1951:2
1947:z
1943:C
1940:+
1935:2
1931:y
1927:B
1924:+
1919:2
1915:x
1911:A
1908:=
1905:)
1902:z
1899:,
1896:y
1893:,
1890:x
1887:(
1884:f
1806:)
1804:r
1801:π
1794:3
1791:/
1788:2
1781:r
1778:π
1776:4
1772:)
1770:r
1767:π
1760:3
1757:/
1754:2
1747:r
1744:π
1738:3
1735:/
1732:4
1724:r
1651:.
1647:)
1641:2
1637:r
1628:2
1624:R
1619:(
1612:2
1609:+
1606:h
1602:)
1598:r
1595:+
1592:R
1588:(
1581:2
1578:=
1575:A
1552:π
1550:2
1536:.
1533:)
1530:r
1524:R
1521:(
1518:h
1514:)
1509:2
1505:r
1502:+
1499:R
1493:(
1486:2
1483:=
1480:h
1476:)
1470:2
1466:r
1457:2
1453:R
1448:(
1441:=
1438:V
1427:R
1421:r
1415:h
1377:p
1373:e
1359:,
1356:p
1350:e
1347:=
1344:L
1334:L
1326:r
1322:h
1316:r
1312:h
1299:r
1295:d
1280:)
1277:h
1274:+
1271:r
1268:(
1265:d
1259:=
1256:)
1253:r
1250:+
1247:h
1244:(
1241:r
1235:2
1232:=
1227:2
1223:r
1216:2
1213:+
1210:h
1207:r
1201:2
1198:=
1195:B
1192:2
1189:+
1186:L
1183:=
1180:A
1158:h
1155:r
1149:2
1146:=
1143:L
1125:L
1113:B
1091:r
1089:π
1082:r
1080:π
1068:h
1063:r
1037:.
1034:h
1028:2
1024:r
1016:=
1006:z
1003:d
996:d
992:s
989:d
984:s
979:r
974:0
961:2
956:0
946:h
941:0
933:=
926:V
896:.
893:h
890:b
887:a
881:=
878:x
875:d
870:h
865:0
857:b
854:a
848:=
845:x
842:d
839:b
836:a
828:h
823:0
815:=
812:x
809:d
806:)
803:x
800:(
797:A
792:h
787:0
779:=
776:V
765:A
761:x
759:(
757:A
752:x
744:π
739:A
730:V
725:h
721:b
717:a
706:h
700:b
694:a
663:h
658:2
654:r
647:=
644:V
630:h
625:r
596:.
580:r
575:=
568:a
561:,
549:=
542:e
527:α
521:r
515:a
506:e
376:h
370:r
143:)
137:(
76:2
50:r
46:d
42:h
23:.
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