2835:
57:
710:
3082:
3065:
823:. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of
2882:
2335:
3213:. Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume.
685:
plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs
605:
2597:
2196:
2010:
3060:{\displaystyle {\begin{array}{rlll}{\text{Sphere surface area}}&=4\pi r^{2}&&=(2\pi r^{2})\times 2\\{\text{Cylinder surface area}}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3\end{array}}}
1695:
2503:
3163:, allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss.
1448:
446:
1546:
675:
1844:
324:
1299:
3432:
Paumard, Patrick; Vaillier, Jacques; Coulary, Bénédicte; Schaeffer, Jacques; Soubannier, Vincent; Mueller, David M.; Brèthes, Daniel; di Rago, Jean-Paul; Velours, Jean (1 February 2002).
337:
surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the
2221:
1895:
1044:
968:
1182:
2071:
2391:
3174:
imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across the
2750:
2664:
1350:
200:
1763:
485:
2779:
2693:
2627:
1231:
900:
1111:
493:
3320:
804:
of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by
2820:
2799:
2713:
808:
that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as the
794:
774:
754:
734:
2518:
101:
of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of
3151:
to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains
3352:
2098:
1921:
846:
827:. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in
3115:, while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.
1570:
3541:
3371:
3124:
3128:
1376:
2410:
354:
3295:
815:
Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by
1472:
3707:
622:
3167:
1787:
30:
3702:
3327:
452:
3651:
247:
2330:{\displaystyle lb+l{\sqrt {\left({\frac {b}{2}}\right)^{2}+h^{2}}}+b{\sqrt {\left({\frac {l}{2}}\right)^{2}+h^{2}}}}
148:, which studies various notions of surface area for irregular objects of any dimension. An important example is the
3620:
3143:
The surface area of an organism is important in several considerations, such as regulation of body temperature and
1248:
828:
145:
1850:
3090:
995:
919:
1138:
122:
2403:
1563:
703:
334:
98:
2038:
2362:
213:
that satisfies several natural requirements. The most fundamental property of the surface area is its
3720:
3132:
2719:
2633:
341:. These properties uniquely characterize surface area for a wide class of geometric surfaces called
2855:
1465:
1322:
170:
2834:
1724:
3679:
2511:
2031:
1717:
796:
tend to infinity doesn't converge. In particular it doesn't converge to the area of the cylinder.
458:
346:
210:
160:
While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical
130:
126:
118:
2755:
2669:
2603:
1203:
3367:
113:), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a
3671:
3632:
3601:
3537:
3502:
3463:
3414:
3346:
3264:
3104:
3096:
832:
820:
149:
141:
3663:
3529:
3494:
3453:
3445:
3433:
3406:
3243:
3222:
1077:
988:
875:
600:{\displaystyle A(S_{D})=\iint _{D}\left|{\vec {r}}_{u}\times {\vec {r}}_{v}\right|\,du\,dv.}
3619:
Stocks, Jodie M.; Taylor, Nigel A.S.; Tipton, Michael J.; Greenleaf, John E. (1 May 2004).
1084:
56:
3697:
3288:
1780:
809:
805:
713:
338:
110:
3589:
2887:
3533:
3238:
3227:
3171:
2805:
2784:
2698:
2091:
1362:
1315:
816:
779:
759:
739:
719:
137:
90:
3458:
3178:
to interstitial spaces or to other cells. Indeed, representing a cell as an idealized
2592:{\displaystyle \iint _{D}\left\vert {\vec {r}}_{u}\times {\vec {r}}_{v}\right\vert dA}
3736:
3730:
3521:
3175:
3100:
329:
Surface areas of flat polygonal shapes must agree with their geometrically defined
3563:
3267:
2191:{\displaystyle b^{2}+2bs=b^{2}+2b{\sqrt {\left({\frac {b}{2}}\right)^{2}+h^{2}}}}
3123:
2355:
206:
2005:{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)=\pi r\left(r+s\right)}
345:. Such surfaces consist of finitely many pieces that can be represented in the
3498:
3232:
3152:
3136:
3112:
3070:
801:
709:
102:
3675:
3605:
3506:
3449:
3434:"The ATP synthase is involved in generating mitochondrial cristae morphology"
3418:
3394:
2830:
Ratio of surface areas of a sphere and cylinder of the same radius and height
3482:
3272:
3144:
106:
3636:
3467:
3395:"Kinetics of Iron Ore Reduction by Methane for Chemical Looping Combustion"
144:
at the turn of the twentieth century. Their work led to the development of
3081:
3235:, technique for the measurement of the specific surface area of materials
3156:
2850:
The below given formulas can be used to show that the surface area of a
1690:{\displaystyle 2\pi Rh+2\pi rh+2(\pi R^{2}-\pi r^{2})=2\pi (R+r)(R-r+h)}
3683:
824:
3410:
164:
of area requires a great deal of care. This should provide a function
3179:
3131:
has a large surface area due to infoldings, allowing higher rates of
3099:. Increasing the surface area of a substance generally increases the
2851:
1131:
912:
114:
61:
3667:
3148:
3122:
3080:
1443:{\displaystyle \left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr}
1369:
708:
55:
3564:"Microvillus | Description, Anatomy, & Function | Britannica"
2498:{\displaystyle 2\pi \int _{a}^{b}{f(x){\sqrt {1+(f'(x))^{2}}}dx}}
1459:= major radius (distance from center of tube to center of torus)
441:{\displaystyle S_{D}:{\vec {r}}={\vec {r}}(u,v),\quad (u,v)\in D}
3108:
868:
330:
94:
3160:
619:
is obtained by integrating the length of the normal vector
3206:. The resulting surface area to volume ratio is therefore
487:
The area of an individual piece is defined by the formula
219:
the area of the whole is the sum of the areas of the parts
117:, are assigned surface area using their representation as
1541:{\displaystyle 2\pi r^{2}+2\pi rh=2\pi r\left(r+h\right)}
121:. This definition of surface area is based on methods of
3696:
Yu.D. Burago; V.A. Zalgaller; L.D. Kudryavtsev (2001) ,
3155:, greatly increasing the area available for absorption.
241:
which do not overlap except at their boundaries, then
34:
3393:
Nasr, Somaye; Plucknett, Kevin P. (20 February 2014).
800:
One of the subtleties of surface area, as compared to
105:
of one-dimensional curves, or of the surface area for
2885:
2808:
2787:
2758:
2722:
2701:
2672:
2636:
2606:
2521:
2413:
2365:
2224:
2101:
2041:
1924:
1853:
1790:
1727:
1573:
1475:
1379:
1325:
1251:
1206:
1141:
1087:
998:
922:
878:
782:
762:
742:
722:
625:
496:
461:
357:
333:. Since surface area is a geometric notion, areas of
250:
173:
670:{\displaystyle {\vec {r}}_{u}\times {\vec {r}}_{v}}
136:A general definition of surface area was sought by
3059:
2814:
2793:
2773:
2744:
2707:
2687:
2658:
2621:
2591:
2497:
2385:
2329:
2190:
2065:
2004:
1889:
1838:
1757:
1689:
1540:
1442:
1344:
1293:
1225:
1176:
1105:
1038:
962:
894:
788:
768:
748:
728:
669:
599:
479:
440:
318:
194:
3186:, the volume and surface area are, respectively,
1839:{\displaystyle \pi r{\sqrt {r^{2}+h^{2}}}=\pi rs}
831:. A specific example of such an extension is the
3621:"Human Physiological Responses to Cold Exposure"
2858:of the same radius and height are in the ratio
319:{\displaystyle A(S)=A(S_{1})+\cdots +A(S_{r}).}
3370:. Courant Institute of Mathematical Sciences.
3085:Surface area of particles of different sizes.
1294:{\displaystyle \pi \left(3R^{2}+r^{2}\right)}
8:
3625:Aviation, Space, and Environmental Medicine
3069:The discovery of this ratio is credited to
677:to the surface over the appropriate region
18:
1770:= radius of the hemispheres and cylinder,
756:radial vertices. The limit of the area as
3457:
3038:
2954:
2935:
2910:
2890:
2886:
2884:
2807:
2786:
2760:
2759:
2757:
2736:
2725:
2724:
2721:
2700:
2674:
2673:
2671:
2650:
2639:
2638:
2635:
2629:= parametric vector equation of surface,
2608:
2607:
2605:
2572:
2561:
2560:
2550:
2539:
2538:
2526:
2520:
2480:
2448:
2435:
2429:
2424:
2412:
2377:
2366:
2364:
2319:
2306:
2292:
2285:
2271:
2258:
2244:
2237:
2223:
2180:
2167:
2153:
2146:
2131:
2106:
2100:
2048:
2040:
1961:
1948:
1942:
1923:
1879:
1866:
1860:
1852:
1816:
1803:
1797:
1789:
1726:
1633:
1617:
1572:
1486:
1474:
1428:
1378:
1333:
1324:
1280:
1267:
1250:
1217:
1205:
1168:
1152:
1140:
1086:
997:
921:
886:
877:
781:
761:
741:
721:
661:
650:
649:
639:
628:
627:
624:
587:
580:
569:
558:
557:
547:
536:
535:
523:
507:
495:
463:
462:
460:
387:
386:
372:
371:
362:
356:
304:
276:
249:
172:
2833:
850:
24:This is an accepted version of this page
3255:
1890:{\displaystyle s={\sqrt {r^{2}+h^{2}}}}
847:List of formulas in elementary geometry
20:
3483:"Why do elephants have big ear flaps?"
3351:: CS1 maint: archived copy as title (
3344:
2838:A cone, sphere and cylinder of radius
1039:{\displaystyle bh+l\left(p+q+r\right)}
963:{\displaystyle 2\left(lb+lh+bh\right)}
1455:= minor radius (radius of the tube),
1059:= distance between triangular bases,
7:
3650:Deaver, James R. (1 November 1978).
3374:from the original on 9 December 2006
1177:{\displaystyle 4\pi r^{2}=\pi d^{2}}
51:Measure of a two-dimensional surface
3590:"Why do elephants flap their ears?"
3129:inner membrane of the mitochondrion
225:is a union of finitely many pieces
109:(i.e., objects with flat polygonal
3534:10.1016/b978-0-12-382163-8.00077-3
49:
3481:Narasimhan, Arunn (1 July 2008).
3301:from the original on 4 March 2016
2066:{\displaystyle B+{\frac {Ps}{2}}}
1306:= external radius of hemisphere,
93:object is a measure of the total
2386:{\displaystyle {\sqrt {3}}a^{2}}
1310:= internal radius of hemisphere
221:. More rigorously, if a surface
2021:= radius of the circular base,
1906:= radius of the circular base,
1553:= radius of the circular base,
852:Surface areas of common solids
416:
3652:"Modeling Limits to Cell Size"
3528:, Elsevier, pp. 689–700,
3044:
3025:
3017:
3002:
2985:
2973:
2941:
2922:
2765:
2745:{\displaystyle {\vec {r}}_{v}}
2730:
2679:
2659:{\displaystyle {\vec {r}}_{u}}
2644:
2613:
2566:
2544:
2477:
2473:
2467:
2456:
2445:
2439:
1752:
1737:
1684:
1666:
1663:
1651:
1639:
1607:
655:
633:
563:
541:
513:
500:
468:
429:
417:
410:
398:
392:
377:
310:
297:
282:
269:
260:
254:
189:
183:
177:
1:
3526:Quantitative Human Physiology
3368:"Tomb of Archimedes: Sources"
3095:Surface area is important in
1345:{\displaystyle 2r^{2}\theta }
1122:= the perimeter of one base,
195:{\displaystyle S\mapsto A(S)}
3656:The American Biology Teacher
3168:surface area to volume ratio
2017:= slant height of the cone,
1916:Full surface area of a cone
1902:= slant height of the cone,
1758:{\displaystyle 2\pi r(2r+h)}
3703:Encyclopedia of Mathematics
1238:= radius of the hemisphere
1051:= base length of triangle,
480:{\displaystyle {\vec {r}}.}
453:continuously differentiable
3753:
3088:
2774:{\displaystyle {\vec {r}}}
2688:{\displaystyle {\vec {r}}}
2622:{\displaystyle {\vec {r}}}
1226:{\displaystyle 3\pi r^{2}}
844:
339:group of Euclidean motions
3499:10.1007/s12045-008-0070-5
1779:Curved surface area of a
1774:= height of the cylinder
1557:= height of the cylinder
205:which assigns a positive
2752:= partial derivative of
2666:= partial derivative of
1118:= the area of one base,
829:geometric measure theory
146:geometric measure theory
31:latest accepted revision
3091:Accessible surface area
3588:Wright, P. G. (1984).
3520:Feher, Joseph (2012),
3450:10.1093/emboj/21.3.221
3140:
3111:in a fine powder will
3086:
3061:
2847:
2816:
2795:
2775:
2746:
2709:
2689:
2660:
2623:
2593:
2499:
2387:
2331:
2192:
2067:
2006:
1891:
1840:
1759:
1691:
1542:
1444:
1346:
1295:
1227:
1178:
1107:
1055:= height of triangle,
1040:
964:
896:
895:{\displaystyle 6a^{2}}
797:
790:
770:
750:
730:
704:surfaces of revolution
671:
601:
481:
442:
320:
209:to a certain class of
196:
123:infinitesimal calculus
76:
3522:"Mouth and Esophagus"
3126:
3084:
3062:
2956:Cylinder surface area
2837:
2817:
2796:
2776:
2747:
2710:
2690:
2661:
2624:
2594:
2500:
2404:Surface of revolution
2388:
2332:
2193:
2082:= perimeter of base,
2068:
2025:= height of the cone
2007:
1910:= height of the cone
1892:
1841:
1760:
1692:
1543:
1445:
1347:
1296:
1228:
1179:
1108:
1106:{\displaystyle 2B+Ph}
1041:
965:
897:
791:
771:
751:
731:
712:
672:
602:
482:
443:
321:
197:
59:
3147:. Animals use their
3133:cellular respiration
2883:
2806:
2785:
2756:
2720:
2699:
2670:
2634:
2604:
2519:
2411:
2363:
2222:
2216:Rectangular pyramid
2099:
2039:
1922:
1851:
1788:
1725:
1571:
1473:
1377:
1357:= radius of sphere,
1323:
1249:
1243:Hemispherical shell
1204:
1189:= radius of sphere,
1139:
1085:
1071:= sides of triangle
996:
920:
876:
780:
760:
740:
720:
623:
494:
459:
355:
248:
171:
3333:on 15 December 2011
3289:"Schwarz's Paradox"
2892:Sphere surface area
2434:
1707:= Internal radius,
853:
127:partial derivatives
119:parametric surfaces
21:Page version status
3721:Surface Area Video
3568:www.britannica.com
3399:Energy & Fuels
3265:Weisstein, Eric W.
3141:
3087:
3057:
3055:
2869:and the height be
2865:Let the radius be
2848:
2812:
2791:
2771:
2742:
2705:
2685:
2656:
2619:
2589:
2512:Parametric surface
2495:
2420:
2383:
2327:
2211:= vertical height
2188:
2063:
2002:
1887:
1836:
1755:
1702:= External radius
1687:
1538:
1440:
1342:
1291:
1223:
1174:
1103:
1036:
960:
892:
851:
798:
786:
766:
746:
726:
681:in the parametric
667:
597:
477:
438:
316:
192:
131:double integration
77:
27:
3543:978-0-12-382163-8
3411:10.1021/ef402142q
3105:chemical reaction
3097:chemical kinetics
2957:
2893:
2877:for the sphere).
2827:
2826:
2815:{\displaystyle D}
2794:{\displaystyle v}
2768:
2733:
2708:{\displaystyle u}
2682:
2647:
2616:
2569:
2547:
2486:
2371:
2325:
2300:
2277:
2252:
2186:
2161:
2061:
1967:
1885:
1822:
860:Formula/Equation
833:Minkowski content
821:Hermann Minkowski
789:{\displaystyle N}
769:{\displaystyle M}
749:{\displaystyle N}
736:axial slices and
729:{\displaystyle M}
658:
636:
610:Thus the area of
566:
544:
471:
395:
380:
150:Minkowski content
142:Hermann Minkowski
68:has surface area
39:17 September 2024
3744:
3710:
3688:
3687:
3647:
3641:
3640:
3616:
3610:
3609:
3585:
3579:
3578:
3576:
3574:
3560:
3554:
3553:
3552:
3550:
3517:
3511:
3510:
3478:
3472:
3471:
3461:
3438:The EMBO Journal
3429:
3423:
3422:
3405:(2): 1387–1395.
3390:
3384:
3383:
3381:
3379:
3363:
3357:
3356:
3350:
3342:
3340:
3338:
3332:
3326:. Archived from
3325:
3317:
3311:
3310:
3308:
3306:
3300:
3293:
3285:
3279:
3278:
3277:
3260:
3244:Surface integral
3223:Perimeter length
3212:
3205:
3195:
3185:
3066:
3064:
3063:
3058:
3056:
3043:
3042:
2958:
2955:
2940:
2939:
2917:
2915:
2914:
2894:
2891:
2822:= shadow region
2821:
2819:
2818:
2813:
2800:
2798:
2797:
2792:
2781:with respect to
2780:
2778:
2777:
2772:
2770:
2769:
2761:
2751:
2749:
2748:
2743:
2741:
2740:
2735:
2734:
2726:
2714:
2712:
2711:
2706:
2695:with respect to
2694:
2692:
2691:
2686:
2684:
2683:
2675:
2665:
2663:
2662:
2657:
2655:
2654:
2649:
2648:
2640:
2628:
2626:
2625:
2620:
2618:
2617:
2609:
2598:
2596:
2595:
2590:
2582:
2578:
2577:
2576:
2571:
2570:
2562:
2555:
2554:
2549:
2548:
2540:
2531:
2530:
2504:
2502:
2501:
2496:
2494:
2487:
2485:
2484:
2466:
2449:
2433:
2428:
2392:
2390:
2389:
2384:
2382:
2381:
2372:
2367:
2336:
2334:
2333:
2328:
2326:
2324:
2323:
2311:
2310:
2305:
2301:
2293:
2286:
2278:
2276:
2275:
2263:
2262:
2257:
2253:
2245:
2238:
2207:= slant height,
2197:
2195:
2194:
2189:
2187:
2185:
2184:
2172:
2171:
2166:
2162:
2154:
2147:
2136:
2135:
2111:
2110:
2078:= area of base,
2072:
2070:
2069:
2064:
2062:
2057:
2049:
2011:
2009:
2008:
2003:
2001:
1997:
1973:
1969:
1968:
1966:
1965:
1953:
1952:
1943:
1896:
1894:
1893:
1888:
1886:
1884:
1883:
1871:
1870:
1861:
1845:
1843:
1842:
1837:
1823:
1821:
1820:
1808:
1807:
1798:
1764:
1762:
1761:
1756:
1696:
1694:
1693:
1688:
1638:
1637:
1622:
1621:
1547:
1545:
1544:
1539:
1537:
1533:
1491:
1490:
1449:
1447:
1446:
1441:
1433:
1432:
1417:
1413:
1398:
1394:
1351:
1349:
1348:
1343:
1338:
1337:
1300:
1298:
1297:
1292:
1290:
1286:
1285:
1284:
1272:
1271:
1232:
1230:
1229:
1224:
1222:
1221:
1183:
1181:
1180:
1175:
1173:
1172:
1157:
1156:
1112:
1110:
1109:
1104:
1045:
1043:
1042:
1037:
1035:
1031:
989:Triangular prism
969:
967:
966:
961:
959:
955:
901:
899:
898:
893:
891:
890:
854:
835:of the surface.
795:
793:
792:
787:
775:
773:
772:
767:
755:
753:
752:
747:
735:
733:
732:
727:
676:
674:
673:
668:
666:
665:
660:
659:
651:
644:
643:
638:
637:
629:
606:
604:
603:
598:
579:
575:
574:
573:
568:
567:
559:
552:
551:
546:
545:
537:
528:
527:
512:
511:
486:
484:
483:
478:
473:
472:
464:
447:
445:
444:
439:
397:
396:
388:
382:
381:
373:
367:
366:
343:piecewise smooth
325:
323:
322:
317:
309:
308:
281:
280:
201:
199:
198:
193:
74:
67:
3752:
3751:
3747:
3746:
3745:
3743:
3742:
3741:
3727:
3726:
3717:
3695:
3692:
3691:
3668:10.2307/4446369
3649:
3648:
3644:
3618:
3617:
3613:
3594:African Zoology
3587:
3586:
3582:
3572:
3570:
3562:
3561:
3557:
3548:
3546:
3544:
3519:
3518:
3514:
3480:
3479:
3475:
3431:
3430:
3426:
3392:
3391:
3387:
3377:
3375:
3366:Rorres, Chris.
3365:
3364:
3360:
3343:
3336:
3334:
3330:
3323:
3321:"Archived copy"
3319:
3318:
3314:
3304:
3302:
3298:
3291:
3287:
3286:
3282:
3263:
3262:
3261:
3257:
3252:
3219:
3207:
3197:
3187:
3183:
3121:
3107:. For example,
3093:
3079:
3054:
3053:
3034:
3020:
2988:
2959:
2951:
2950:
2931:
2916:
2906:
2895:
2881:
2880:
2860:2 : 3
2832:
2804:
2803:
2802:
2783:
2782:
2754:
2753:
2723:
2718:
2717:
2716:
2697:
2696:
2668:
2667:
2637:
2632:
2631:
2602:
2601:
2559:
2537:
2536:
2532:
2522:
2517:
2516:
2476:
2459:
2409:
2408:
2373:
2361:
2360:
2315:
2288:
2287:
2267:
2240:
2239:
2220:
2219:
2203:= base length,
2176:
2149:
2148:
2127:
2102:
2097:
2096:
2086:= slant height
2050:
2037:
2036:
1987:
1983:
1957:
1944:
1935:
1931:
1920:
1919:
1897:
1875:
1862:
1849:
1848:
1812:
1799:
1786:
1785:
1723:
1722:
1629:
1613:
1569:
1568:
1523:
1519:
1482:
1471:
1470:
1424:
1403:
1399:
1384:
1380:
1375:
1374:
1329:
1321:
1320:
1276:
1263:
1259:
1255:
1247:
1246:
1213:
1202:
1201:
1164:
1148:
1137:
1136:
1083:
1082:
1015:
1011:
994:
993:
930:
926:
918:
917:
882:
874:
873:
849:
841:
839:Common formulas
810:Schwarz lantern
806:Hermann Schwarz
778:
777:
758:
757:
738:
737:
718:
717:
714:Schwarz lantern
648:
626:
621:
620:
618:
556:
534:
533:
529:
519:
503:
492:
491:
457:
456:
358:
353:
352:
347:parametric form
300:
272:
246:
245:
240:
231:
169:
168:
158:
69:
65:
52:
47:
46:
45:
44:
43:
42:
26:
12:
11:
5:
3750:
3748:
3740:
3739:
3729:
3728:
3725:
3724:
3716:
3715:External links
3713:
3712:
3711:
3690:
3689:
3662:(8): 502–504.
3642:
3631:(5): 444–457.
3611:
3600:(4): 266–269.
3580:
3555:
3542:
3512:
3493:(7): 638–647.
3473:
3444:(3): 221–230.
3424:
3385:
3358:
3312:
3280:
3268:"Surface Area"
3254:
3253:
3251:
3248:
3247:
3246:
3241:
3239:Spherical area
3236:
3230:
3228:Projected area
3225:
3218:
3215:
3120:
3117:
3078:
3075:
3052:
3049:
3046:
3041:
3037:
3033:
3030:
3027:
3024:
3021:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2953:
2952:
2949:
2946:
2943:
2938:
2934:
2930:
2927:
2924:
2921:
2918:
2913:
2909:
2905:
2902:
2899:
2896:
2889:
2888:
2862:, as follows.
2831:
2828:
2825:
2824:
2811:
2790:
2767:
2764:
2739:
2732:
2729:
2704:
2681:
2678:
2653:
2646:
2643:
2615:
2612:
2599:
2588:
2585:
2581:
2575:
2568:
2565:
2558:
2553:
2546:
2543:
2535:
2529:
2525:
2514:
2508:
2507:
2505:
2493:
2490:
2483:
2479:
2475:
2472:
2469:
2465:
2462:
2458:
2455:
2452:
2447:
2444:
2441:
2438:
2432:
2427:
2423:
2419:
2416:
2406:
2400:
2399:
2398:= side length
2393:
2380:
2376:
2370:
2358:
2352:
2351:
2337:
2322:
2318:
2314:
2309:
2304:
2299:
2296:
2291:
2284:
2281:
2274:
2270:
2266:
2261:
2256:
2251:
2248:
2243:
2236:
2233:
2230:
2227:
2217:
2213:
2212:
2198:
2183:
2179:
2175:
2170:
2165:
2160:
2157:
2152:
2145:
2142:
2139:
2134:
2130:
2126:
2123:
2120:
2117:
2114:
2109:
2105:
2094:
2092:Square pyramid
2088:
2087:
2073:
2060:
2056:
2053:
2047:
2044:
2034:
2027:
2026:
2012:
2000:
1996:
1993:
1990:
1986:
1982:
1979:
1976:
1972:
1964:
1960:
1956:
1951:
1947:
1941:
1938:
1934:
1930:
1927:
1917:
1913:
1912:
1882:
1878:
1874:
1869:
1865:
1859:
1856:
1846:
1835:
1832:
1829:
1826:
1819:
1815:
1811:
1806:
1802:
1796:
1793:
1783:
1776:
1775:
1765:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1720:
1714:
1713:
1697:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1636:
1632:
1628:
1625:
1620:
1616:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1566:
1559:
1558:
1548:
1536:
1532:
1529:
1526:
1522:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1489:
1485:
1481:
1478:
1468:
1461:
1460:
1450:
1439:
1436:
1431:
1427:
1423:
1420:
1416:
1412:
1409:
1406:
1402:
1397:
1393:
1390:
1387:
1383:
1372:
1366:
1365:
1363:dihedral angle
1352:
1341:
1336:
1332:
1328:
1318:
1316:Spherical lune
1312:
1311:
1301:
1289:
1283:
1279:
1275:
1270:
1266:
1262:
1258:
1254:
1244:
1240:
1239:
1233:
1220:
1216:
1212:
1209:
1199:
1195:
1194:
1184:
1171:
1167:
1163:
1160:
1155:
1151:
1147:
1144:
1134:
1128:
1127:
1113:
1102:
1099:
1096:
1093:
1090:
1080:
1073:
1072:
1046:
1034:
1030:
1027:
1024:
1021:
1018:
1014:
1010:
1007:
1004:
1001:
991:
985:
984:
970:
958:
954:
951:
948:
945:
942:
939:
936:
933:
929:
925:
915:
909:
908:
907:= side length
902:
889:
885:
881:
871:
865:
864:
861:
858:
840:
837:
817:Henri Lebesgue
785:
765:
745:
725:
664:
657:
654:
647:
642:
635:
632:
614:
608:
607:
596:
593:
590:
586:
583:
578:
572:
565:
562:
555:
550:
543:
540:
532:
526:
522:
518:
515:
510:
506:
502:
499:
476:
470:
467:
449:
448:
437:
434:
431:
428:
425:
422:
419:
415:
412:
409:
406:
403:
400:
394:
391:
385:
379:
376:
370:
365:
361:
327:
326:
315:
312:
307:
303:
299:
296:
293:
290:
287:
284:
279:
275:
271:
268:
265:
262:
259:
256:
253:
236:
229:
203:
202:
191:
188:
185:
182:
179:
176:
157:
154:
152:of a surface.
138:Henri Lebesgue
50:
48:
28:
22:
19:
17:
16:
15:
14:
13:
10:
9:
6:
4:
3:
2:
3749:
3738:
3735:
3734:
3732:
3722:
3719:
3718:
3714:
3709:
3705:
3704:
3699:
3694:
3693:
3685:
3681:
3677:
3673:
3669:
3665:
3661:
3657:
3653:
3646:
3643:
3638:
3634:
3630:
3626:
3622:
3615:
3612:
3607:
3603:
3599:
3595:
3591:
3584:
3581:
3569:
3565:
3559:
3556:
3545:
3539:
3535:
3531:
3527:
3523:
3516:
3513:
3508:
3504:
3500:
3496:
3492:
3488:
3484:
3477:
3474:
3469:
3465:
3460:
3455:
3451:
3447:
3443:
3439:
3435:
3428:
3425:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3389:
3386:
3373:
3369:
3362:
3359:
3354:
3348:
3329:
3322:
3316:
3313:
3297:
3290:
3284:
3281:
3275:
3274:
3269:
3266:
3259:
3256:
3249:
3245:
3242:
3240:
3237:
3234:
3231:
3229:
3226:
3224:
3221:
3220:
3216:
3214:
3211:
3204:
3200:
3194:
3190:
3181:
3177:
3176:cell membrane
3173:
3169:
3164:
3162:
3158:
3154:
3150:
3146:
3138:
3134:
3130:
3125:
3118:
3116:
3114:
3110:
3106:
3102:
3098:
3092:
3083:
3076:
3074:
3072:
3067:
3050:
3047:
3039:
3035:
3031:
3028:
3022:
3014:
3011:
3008:
3005:
2999:
2996:
2993:
2990:
2982:
2979:
2976:
2970:
2967:
2964:
2961:
2947:
2944:
2936:
2932:
2928:
2925:
2919:
2911:
2907:
2903:
2900:
2897:
2878:
2876:
2872:
2868:
2863:
2861:
2857:
2853:
2845:
2841:
2836:
2829:
2823:
2809:
2788:
2762:
2737:
2727:
2702:
2676:
2651:
2641:
2610:
2600:
2586:
2583:
2579:
2573:
2563:
2556:
2551:
2541:
2533:
2527:
2523:
2515:
2513:
2510:
2509:
2506:
2491:
2488:
2481:
2470:
2463:
2460:
2453:
2450:
2442:
2436:
2430:
2425:
2421:
2417:
2414:
2407:
2405:
2402:
2401:
2397:
2394:
2378:
2374:
2368:
2359:
2357:
2354:
2353:
2349:
2345:
2341:
2338:
2320:
2316:
2312:
2307:
2302:
2297:
2294:
2289:
2282:
2279:
2272:
2268:
2264:
2259:
2254:
2249:
2246:
2241:
2234:
2231:
2228:
2225:
2218:
2215:
2214:
2210:
2206:
2202:
2199:
2181:
2177:
2173:
2168:
2163:
2158:
2155:
2150:
2143:
2140:
2137:
2132:
2128:
2124:
2121:
2118:
2115:
2112:
2107:
2103:
2095:
2093:
2090:
2089:
2085:
2081:
2077:
2074:
2058:
2054:
2051:
2045:
2042:
2035:
2033:
2029:
2028:
2024:
2020:
2016:
2013:
1998:
1994:
1991:
1988:
1984:
1980:
1977:
1974:
1970:
1962:
1958:
1954:
1949:
1945:
1939:
1936:
1932:
1928:
1925:
1918:
1915:
1914:
1911:
1909:
1905:
1901:
1880:
1876:
1872:
1867:
1863:
1857:
1854:
1847:
1833:
1830:
1827:
1824:
1817:
1813:
1809:
1804:
1800:
1794:
1791:
1784:
1782:
1778:
1777:
1773:
1769:
1766:
1749:
1746:
1743:
1740:
1734:
1731:
1728:
1721:
1719:
1716:
1715:
1712:
1710:
1706:
1701:
1698:
1681:
1678:
1675:
1672:
1669:
1660:
1657:
1654:
1648:
1645:
1642:
1634:
1630:
1626:
1623:
1618:
1614:
1610:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1567:
1565:
1561:
1560:
1556:
1552:
1549:
1534:
1530:
1527:
1524:
1520:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1487:
1483:
1479:
1476:
1469:
1467:
1463:
1462:
1458:
1454:
1451:
1437:
1434:
1429:
1425:
1421:
1418:
1414:
1410:
1407:
1404:
1400:
1395:
1391:
1388:
1385:
1381:
1373:
1371:
1368:
1367:
1364:
1360:
1356:
1353:
1339:
1334:
1330:
1326:
1319:
1317:
1314:
1313:
1309:
1305:
1302:
1287:
1281:
1277:
1273:
1268:
1264:
1260:
1256:
1252:
1245:
1242:
1241:
1237:
1234:
1218:
1214:
1210:
1207:
1200:
1197:
1196:
1192:
1188:
1185:
1169:
1165:
1161:
1158:
1153:
1149:
1145:
1142:
1135:
1133:
1130:
1129:
1125:
1121:
1117:
1114:
1100:
1097:
1094:
1091:
1088:
1081:
1079:
1075:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1047:
1032:
1028:
1025:
1022:
1019:
1016:
1012:
1008:
1005:
1002:
999:
992:
990:
987:
986:
982:
978:
974:
971:
956:
952:
949:
946:
943:
940:
937:
934:
931:
927:
923:
916:
914:
911:
910:
906:
903:
887:
883:
879:
872:
870:
867:
866:
862:
859:
856:
855:
848:
843:
838:
836:
834:
830:
826:
822:
818:
813:
811:
807:
803:
783:
763:
743:
723:
715:
711:
707:
705:
701:
697:
693:
689:
684:
680:
662:
652:
645:
640:
630:
617:
613:
594:
591:
588:
584:
581:
576:
570:
560:
553:
548:
538:
530:
524:
520:
516:
508:
504:
497:
490:
489:
488:
474:
465:
454:
435:
432:
426:
423:
420:
413:
407:
404:
401:
389:
383:
374:
368:
363:
359:
351:
350:
349:
348:
344:
340:
336:
332:
313:
305:
301:
294:
291:
288:
285:
277:
273:
266:
263:
257:
251:
244:
243:
242:
239:
235:
228:
224:
220:
216:
212:
208:
186:
180:
174:
167:
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3068:
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2873:(which is 2
2842:and height
2356:Tetrahedron
2346:= breadth,
1198:Hemisphere
1193:= diameter
979:= breadth,
207:real number
3250:References
3233:BET theory
3182:of radius
3153:microvilli
3137:micrograph
3135:(electron
3119:In biology
3089:See also:
3071:Archimedes
2342:= length,
975:= length,
863:Variables
845:See also:
802:arc length
215:additivity
162:definition
156:Definition
103:arc length
64:of radius
3708:EMS Press
3676:0002-7685
3606:2224-073X
3507:0973-712X
3487:Resonance
3419:0887-0624
3378:2 January
3273:MathWorld
3157:Elephants
3145:digestion
3048:×
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2997:π
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289:⋯
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107:polyhedra
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1466:cylinder
825:fractals
211:surfaces
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3191:= (4/3)
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1078:prisms
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857:Shape
702:) and
115:sphere
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3680:JSTOR
3331:(PDF)
3324:(PDF)
3299:(PDF)
3292:(PDF)
3149:teeth
3103:of a
1370:Torus
716:with
232:, …,
111:faces
91:solid
3737:Area
3672:ISSN
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3602:ISSN
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3503:ISSN
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