2144:
314:
183:
1675:
45:
40:
1699:
2018:
306:
1690:
750:
940:
1646:
1991:
1100:
1482:
1352:
1815:
1250:
1880:
461:
567:
617:
803:
325:
A frustum's axis is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.
1528:
1146:
773:
1889:
1728:
2216:, a different Latin word cognate to the English word "frustrate". The confusion between these two words is very old: a warning about them can be found in the
2478:
2153:
2147:
982:
481:
The
Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.
1380:
1261:
1735:
291:
necessarily parallel to the cone's base, as in a frustum. If all its edges are forced to become of the same length, then a frustum becomes a
2397:
2471:
1176:
1824:
388:
2441:
488:
of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":
2464:
494:
2052:
745:{\displaystyle {\frac {B_{1}}{h_{1}^{2}}}={\frac {B_{2}}{h_{2}^{2}}}={\frac {\sqrt {B_{1}B_{2}}}{h_{1}h_{2}}}=\alpha ,}
2279:
128:
935:{\displaystyle V={\frac {h_{1}\alpha h_{1}^{2}-h_{2}\alpha h_{2}^{2}}{3}}=\alpha {\frac {h_{1}^{3}-h_{2}^{3}}{3}}.}
331:
Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the
2351:
Al-Sammarraie, Ahmed T.; Vafai, Kambiz (2017). "Heat transfer augmentation through convergence angles in a pipe".
2034:
2143:
2885:
2030:
2677:
2618:
776:
368:
2707:
2667:
2702:
2697:
2446:
2055:
mounds also form the frustum of one or more pyramids, with additional features such as stairs added.
1641:{\displaystyle V={\frac {nh}{12}}\left(a_{1}^{2}+a_{1}a_{2}+a_{2}^{2}\right)\cot {\frac {\pi }{n}},}
20:
2808:
2803:
2682:
2588:
2271:
1674:
755:
the formula for the volume can be expressed as the third of the product of this proportionality,
2250:
Teachers' Manual: Books I–VIII. For Prang's complete course in form-study and drawing, Books 7–8
133:
2672:
2613:
2603:
2548:
2265:
2692:
2608:
2563:
2511:
2248:
1986:{\displaystyle \displaystyle \pi \left(\left(r_{1}+r_{2}\right)s+r_{1}^{2}+r_{2}^{2}\right),}
1131:
758:
343:
266:
335:(so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of
2652:
2578:
2526:
2360:
2091:
364:
328:
The height of a frustum is the perpendicular distance between the planes of the two bases.
8:
2818:
2687:
2662:
2647:
2583:
2531:
2080:
2065:
1357:
313:
2364:
2880:
2833:
2798:
2657:
2552:
2501:
2376:
2024:
brand chocolates approximate a right circular conic frustum, although not flat on top.
1713:
363:
The formula for the volume of a pyramidal square frustum was introduced by the ancient
236:
2408:
2159:
2813:
2623:
2598:
2542:
2424:
2405:
2380:
2275:
2177:
2038:
1507:
283:
113:
2427:
2752:
2368:
2102:
2058:
293:
174:
2372:
1095:{\displaystyle V=(h_{1}-h_{2})\alpha {\frac {h_{1}^{2}+h_{1}h_{2}+h_{2}^{2}}{3}},}
2087:
332:
244:
240:
153:
98:
87:
71:
2207:
1477:{\displaystyle V={\frac {\pi h}{3}}\left(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}\right),}
182:
2573:
2496:
2218:
2201:
2118:
1361:
277:
216:
1347:{\displaystyle V={\frac {h}{3}}\left(B_{1}+{\sqrt {B_{1}B_{2}}}+B_{2}\right).}
2874:
2778:
2634:
2568:
2136:
2106:
2095:
1810:{\displaystyle \displaystyle s={\sqrt {\left(r_{1}-r_{2}\right)^{2}+h^{2}}},}
1149:
262:
2111:
2048:
1708:
372:
16:
Portion of a solid that lies between two parallel planes cutting the solid
2330:
252:
2451:
608:
are the perpendicular heights from the apex to the base and top planes.
44:
2843:
2731:
2521:
2488:
2212:, meaning 'piece' or 'morsel". The English word is often misspelled as
336:
318:
232:
2838:
2828:
2773:
2757:
2593:
2432:
2413:
2129:
2083:
is a narrow square-based pyramidal frustum topped by a small pyramid.
347:
248:
159:
80:
39:
2398:
Derivation of formula for the volume of frustums of pyramid and cone
2724:
2456:
2073:
2044:
204:
2017:
1360:
is noted for deriving this formula, and with it, encountering the
2848:
2223:
2069:
1698:
305:
300:
247:
cutting the solid. In the case of a pyramid, the base faces are
2125:
485:
1689:
1245:{\displaystyle {\frac {B_{1}+{\sqrt {B_{1}B_{2}}}+B_{2}}{3}};}
1875:{\displaystyle \displaystyle \pi \left(r_{1}+r_{2}\right)s,}
456:{\displaystyle V={\frac {h}{3}}\left(a^{2}+ab+b^{2}\right),}
2516:
2164:
2021:
2403:
1515:
The volume of a pyramidal frustum whose bases are regular
321:
can be augmented on 3 faces to create a triangular frustum
2452:
Design paper models of conical frustum (truncated cones)
2422:
1893:
1892:
1828:
1827:
1739:
1738:
1716:
1531:
1383:
1264:
1179:
1134:
985:
806:
761:
620:
562:{\displaystyle V={\frac {h_{1}B_{1}-h_{2}B_{2}}{3}},}
497:
391:
2033:, a pyramidal frustum appears on the reverse of the
2094:is a virtual photographic or video camera's usable
2350:
1985:
1874:
1809:
1722:
1640:
1476:
1346:
1244:
1140:
1094:
934:
767:
744:
561:
455:
269:perpendicularly to its axis; otherwise, it is an
2872:
297:(possibly oblique or/and with irregular bases).
2122:claims that "every frustum longs to be a cone".
49:Examples: right pentagonal and square frustums
2472:
2442:Paper models of frustums (truncated pyramids)
2353:Numerical Heat Transfer, Part A: Applications
346:bases joined at these congruent bases make a
301:Elements, special cases, and related concepts
208:
2479:
2465:
2295:Kern, William F.; Bland, James R. (1938).
2132:are everyday examples of conical frustums.
1148:and substituting from its definition, the
181:
2294:
2009:are the base and top radii respectively.
1707:For a right circular conical frustum the
1371:The volume of a circular cone frustum is:
2263:
2253:. Prang Educational Company. p. 49.
2156:wooden structure or statue in Lithuania.
2142:
2076:is a frustum whose bases are rectangles.
2016:
1696:
1688:
312:
304:
190:Example: net of right trigonal frustum (
2447:Paper model of frustum (truncated cone)
474:are the base and top side lengths, and
2873:
1255:the alternative formula is therefore:
2460:
2423:
2404:
2246:
2486:
1364:: the square root of negative one.
220:
13:
1673:
1670:are the base and top side lengths.
14:
2897:
2391:
2320:Princeton University Press. 1998
2311:An Imaginary Tale: The story of
590:are the base and top areas, and
43:
38:
2098:modeled as a pyramidal frustum.
2035:Great Seal of the United States
2029:On the back (the reverse) of a
1697:
1684:
2344:
2323:
2303:
2288:
2267:Funny Words in Plautine Comedy
2257:
2240:
2190:
1884:and the total surface area is
1703:3D model of a conical frustum.
1125:is the height of the frustum.
1018:
992:
1:
2373:10.1080/10407782.2017.1372670
2297:Solid Mensuration with Proofs
2233:
2031:United States one-dollar bill
376:
2859:Degenerate polyhedra are in
2247:Clark, John Spencer (1895).
1819:the lateral surface area is
7:
2678:pentagonal icositetrahedron
2619:truncated icosidodecahedron
2171:
2012:
369:Moscow Mathematical Papyrus
353:
10:
2902:
2708:pentagonal hexecontahedron
2668:deltoidal icositetrahedron
2264:Fontaine, Michael (2010).
2135:Drinking glasses and some
2109:'s short-story collection
287:, the truncation plane is
18:
2857:
2791:
2766:
2748:
2741:
2716:
2703:disdyakis triacontahedron
2698:deltoidal hexecontahedron
2632:
2540:
2495:
358:
209:
189:
180:
173:
165:
152:
127:
112:
97:
70:
37:
28:
2331:"Mathwords.com: Frustum"
2183:
243:) that lies between two
219:for 'morsel'); (
21:Frustum (disambiguation)
2809:gyroelongated bipyramid
2683:rhombic triacontahedron
2589:truncated cuboctahedron
2272:Oxford University Press
2139:are also some examples.
1141:{\displaystyle \alpha }
777:difference of the cubes
768:{\displaystyle \alpha }
251:and the side faces are
29:Set of pyramidal right
2804:truncated trapezohedra
2673:disdyakis dodecahedron
2639:(duals of Archimedean)
2614:rhombicosidodecahedron
2604:truncated dodecahedron
2226:include a pun on them.
2206:
2151:
2051:, and certain ancient
2025:
1987:
1876:
1811:
1724:
1704:
1694:
1679:
1642:
1478:
1348:
1246:
1142:
1096:
945:By using the identity
936:
769:
746:
563:
457:
367:in what is called the
322:
310:
231:) is the portion of a
2693:pentakis dodecahedron
2609:truncated icosahedron
2564:truncated tetrahedron
2274:. pp. 117, 154.
2146:
2020:
1988:
1877:
1812:
1725:
1702:
1692:
1677:
1643:
1506:are the base and top
1479:
1349:
1247:
1143:
1097:
937:
770:
747:
564:
458:
316:
308:
2886:Prismatoid polyhedra
2653:rhombic dodecahedron
2579:truncated octahedron
2150:, Neringa, Lithuania
2092:3D computer graphics
2037:, surmounted by the
1890:
1825:
1736:
1714:
1529:
1381:
1262:
1177:
1132:
983:
804:
759:
618:
495:
389:
365:Egyptian mathematics
342:Two frusta with two
19:For other uses, see
2688:triakis icosahedron
2663:tetrakis hexahedron
2648:triakis tetrahedron
2584:rhombicuboctahedron
2409:"Pyramidal frustum"
2365:2017NHTA...72..197A
2222:, and the works of
2081:Washington Monument
2066:John Hancock Center
1973:
1955:
1613:
1572:
1465:
1424:
1358:Heron of Alexandria
1082:
1041:
922:
904:
874:
843:
679:
647:
2658:triakis octahedron
2543:Archimedean solids
2425:Weisstein, Eric W.
2406:Weisstein, Eric W.
2152:
2026:
1983:
1982:
1959:
1941:
1872:
1871:
1807:
1806:
1720:
1705:
1695:
1680:
1638:
1599:
1558:
1474:
1451:
1410:
1344:
1242:
1138:
1092:
1068:
1027:
932:
908:
890:
860:
829:
765:
742:
665:
633:
559:
453:
323:
311:
2868:
2867:
2787:
2786:
2624:snub dodecahedron
2599:icosidodecahedron
2428:"Conical frustum"
2178:Spherical frustum
2039:Eye of Providence
1801:
1723:{\displaystyle s}
1678:Pyramidal frustum
1633:
1551:
1403:
1321:
1279:
1237:
1218:
1087:
927:
879:
731:
708:
680:
648:
611:Considering that
554:
406:
371:, written in the
284:truncated pyramid
201:
200:
2893:
2746:
2745:
2742:Dihedral uniform
2717:Dihedral regular
2640:
2556:
2505:
2481:
2474:
2467:
2458:
2457:
2438:
2437:
2419:
2418:
2385:
2384:
2348:
2342:
2341:
2339:
2337:
2327:
2321:
2317:
2316:
2307:
2301:
2300:
2292:
2286:
2285:
2261:
2255:
2254:
2244:
2227:
2194:
2059:Chinese pyramids
1992:
1990:
1989:
1984:
1978:
1974:
1972:
1967:
1954:
1949:
1934:
1930:
1929:
1928:
1916:
1915:
1881:
1879:
1878:
1873:
1864:
1860:
1859:
1858:
1846:
1845:
1816:
1814:
1813:
1808:
1802:
1800:
1799:
1787:
1786:
1781:
1777:
1776:
1775:
1763:
1762:
1747:
1729:
1727:
1726:
1721:
1701:
1669:
1660:
1647:
1645:
1644:
1639:
1634:
1626:
1618:
1614:
1612:
1607:
1595:
1594:
1585:
1584:
1571:
1566:
1552:
1547:
1539:
1518:
1505:
1496:
1483:
1481:
1480:
1475:
1470:
1466:
1464:
1459:
1447:
1446:
1437:
1436:
1423:
1418:
1404:
1399:
1391:
1353:
1351:
1350:
1345:
1340:
1336:
1335:
1334:
1322:
1320:
1319:
1310:
1309:
1300:
1295:
1294:
1280:
1272:
1251:
1249:
1248:
1243:
1238:
1233:
1232:
1231:
1219:
1217:
1216:
1207:
1206:
1197:
1192:
1191:
1181:
1169:
1160:
1147:
1145:
1144:
1139:
1124:
1101:
1099:
1098:
1093:
1088:
1083:
1081:
1076:
1064:
1063:
1054:
1053:
1040:
1035:
1025:
1017:
1016:
1004:
1003:
975:
941:
939:
938:
933:
928:
923:
921:
916:
903:
898:
888:
880:
875:
873:
868:
856:
855:
842:
837:
825:
824:
814:
796:
787:
774:
772:
771:
766:
751:
749:
748:
743:
732:
730:
729:
728:
719:
718:
707:
706:
697:
696:
687:
686:
681:
678:
673:
664:
663:
654:
649:
646:
641:
632:
631:
622:
607:
598:
589:
580:
568:
566:
565:
560:
555:
550:
549:
548:
539:
538:
526:
525:
516:
515:
505:
477:
473:
469:
462:
460:
459:
454:
449:
445:
444:
443:
422:
421:
407:
399:
381:
378:
265:or a right cone
222:
214:
213:
196:
185:
147:
123:
108:
91:
86:
78:
64:
57:
47:
42:
32:
26:
25:
2901:
2900:
2896:
2895:
2894:
2892:
2891:
2890:
2871:
2870:
2869:
2864:
2853:
2792:Dihedral others
2783:
2762:
2737:
2712:
2641:
2638:
2637:
2628:
2557:
2546:
2545:
2536:
2499:
2497:Platonic solids
2491:
2485:
2400:(Mathalino.com)
2394:
2389:
2388:
2349:
2345:
2335:
2333:
2329:
2328:
2324:
2314:
2312:
2308:
2304:
2293:
2289:
2282:
2262:
2258:
2245:
2241:
2236:
2231:
2230:
2200:comes from
2195:
2191:
2186:
2174:
2160:Valençay cheese
2105:translation of
2088:viewing frustum
2053:Native American
2015:
2008:
2001:
1993:
1968:
1963:
1950:
1945:
1924:
1920:
1911:
1907:
1906:
1902:
1901:
1897:
1891:
1888:
1887:
1882:
1854:
1850:
1841:
1837:
1836:
1832:
1826:
1823:
1822:
1817:
1795:
1791:
1782:
1771:
1767:
1758:
1754:
1753:
1749:
1748:
1746:
1737:
1734:
1733:
1715:
1712:
1711:
1693:Conical frustum
1687:
1668:
1662:
1659:
1653:
1625:
1608:
1603:
1590:
1586:
1580:
1576:
1567:
1562:
1557:
1553:
1540:
1538:
1530:
1527:
1526:
1516:
1504:
1498:
1495:
1489:
1460:
1455:
1442:
1438:
1432:
1428:
1419:
1414:
1409:
1405:
1392:
1390:
1382:
1379:
1378:
1367:In particular:
1330:
1326:
1315:
1311:
1305:
1301:
1299:
1290:
1286:
1285:
1281:
1271:
1263:
1260:
1259:
1227:
1223:
1212:
1208:
1202:
1198:
1196:
1187:
1183:
1182:
1180:
1178:
1175:
1174:
1168:
1162:
1159:
1153:
1133:
1130:
1129:
1119:
1112:
1106:
1077:
1072:
1059:
1055:
1049:
1045:
1036:
1031:
1026:
1024:
1012:
1008:
999:
995:
984:
981:
980:
946:
917:
912:
899:
894:
889:
887:
869:
864:
851:
847:
838:
833:
820:
816:
815:
813:
805:
802:
801:
795:
789:
786:
780:
779:of the heights
760:
757:
756:
724:
720:
714:
710:
709:
702:
698:
692:
688:
685:
674:
669:
659:
655:
653:
642:
637:
627:
623:
621:
619:
616:
615:
606:
600:
597:
591:
588:
582:
579:
573:
544:
540:
534:
530:
521:
517:
511:
507:
506:
504:
496:
493:
492:
478:is the height.
475:
471:
467:
439:
435:
417:
413:
412:
408:
398:
390:
387:
386:
379:
361:
356:
303:
272:oblique frustum
245:parallel planes
191:
154:Dual polyhedron
141:
134:
118:
103:
89:
84:
76:
66:
59:
52:
50:
33:-gonal frustums
30:
24:
17:
12:
11:
5:
2899:
2889:
2888:
2883:
2866:
2865:
2858:
2855:
2854:
2852:
2851:
2846:
2841:
2836:
2831:
2826:
2821:
2816:
2811:
2806:
2801:
2795:
2793:
2789:
2788:
2785:
2784:
2782:
2781:
2776:
2770:
2768:
2764:
2763:
2761:
2760:
2755:
2749:
2743:
2739:
2738:
2736:
2735:
2728:
2720:
2718:
2714:
2713:
2711:
2710:
2705:
2700:
2695:
2690:
2685:
2680:
2675:
2670:
2665:
2660:
2655:
2650:
2644:
2642:
2635:Catalan solids
2633:
2630:
2629:
2627:
2626:
2621:
2616:
2611:
2606:
2601:
2596:
2591:
2586:
2581:
2576:
2574:truncated cube
2571:
2566:
2560:
2558:
2541:
2538:
2537:
2535:
2534:
2529:
2524:
2519:
2514:
2508:
2506:
2493:
2492:
2484:
2483:
2476:
2469:
2461:
2455:
2454:
2449:
2444:
2439:
2420:
2401:
2393:
2392:External links
2390:
2387:
2386:
2359:(3): 197−214.
2343:
2322:
2302:
2287:
2280:
2256:
2238:
2237:
2235:
2232:
2229:
2228:
2219:Appendix Probi
2188:
2187:
2185:
2182:
2181:
2180:
2173:
2170:
2169:
2168:
2162:
2157:
2154:Garsų Gaudyklė
2148:Garsų Gaudyklė
2140:
2137:space capsules
2133:
2123:
2119:tensor algebra
2099:
2084:
2077:
2062:
2056:
2042:
2014:
2011:
2006:
1999:
1981:
1977:
1971:
1966:
1962:
1958:
1953:
1948:
1944:
1940:
1937:
1933:
1927:
1923:
1919:
1914:
1910:
1905:
1900:
1896:
1886:
1870:
1867:
1863:
1857:
1853:
1849:
1844:
1840:
1835:
1831:
1821:
1805:
1798:
1794:
1790:
1785:
1780:
1774:
1770:
1766:
1761:
1757:
1752:
1745:
1742:
1732:
1719:
1686:
1683:
1682:
1681:
1671:
1666:
1657:
1650:
1649:
1648:
1637:
1632:
1629:
1624:
1621:
1617:
1611:
1606:
1602:
1598:
1593:
1589:
1583:
1579:
1575:
1570:
1565:
1561:
1556:
1550:
1546:
1543:
1537:
1534:
1521:
1520:
1512:
1511:
1502:
1493:
1486:
1485:
1484:
1473:
1469:
1463:
1458:
1454:
1450:
1445:
1441:
1435:
1431:
1427:
1422:
1417:
1413:
1408:
1402:
1398:
1395:
1389:
1386:
1373:
1372:
1362:imaginary unit
1355:
1354:
1343:
1339:
1333:
1329:
1325:
1318:
1314:
1308:
1304:
1298:
1293:
1289:
1284:
1278:
1275:
1270:
1267:
1253:
1252:
1241:
1236:
1230:
1226:
1222:
1215:
1211:
1205:
1201:
1195:
1190:
1186:
1166:
1157:
1137:
1117:
1110:
1103:
1102:
1091:
1086:
1080:
1075:
1071:
1067:
1062:
1058:
1052:
1048:
1044:
1039:
1034:
1030:
1023:
1020:
1015:
1011:
1007:
1002:
998:
994:
991:
988:
943:
942:
931:
926:
920:
915:
911:
907:
902:
897:
893:
886:
883:
878:
872:
867:
863:
859:
854:
850:
846:
841:
836:
832:
828:
823:
819:
812:
809:
793:
784:
764:
753:
752:
741:
738:
735:
727:
723:
717:
713:
705:
701:
695:
691:
684:
677:
672:
668:
662:
658:
652:
645:
640:
636:
630:
626:
604:
595:
586:
577:
570:
569:
558:
553:
547:
543:
537:
533:
529:
524:
520:
514:
510:
503:
500:
464:
463:
452:
448:
442:
438:
434:
431:
428:
425:
420:
416:
411:
405:
402:
397:
394:
380: 1850 BC
360:
357:
355:
352:
309:Square frustum
302:
299:
290:
278:truncated cone
199:
198:
187:
186:
178:
177:
171:
170:
167:
163:
162:
156:
150:
149:
136:
131:
129:Symmetry group
125:
124:
116:
110:
109:
101:
95:
94:
74:
68:
67:
48:
35:
34:
15:
9:
6:
4:
3:
2:
2898:
2887:
2884:
2882:
2879:
2878:
2876:
2862:
2856:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2796:
2794:
2790:
2780:
2777:
2775:
2772:
2771:
2769:
2765:
2759:
2756:
2754:
2751:
2750:
2747:
2744:
2740:
2734:
2733:
2729:
2727:
2726:
2722:
2721:
2719:
2715:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2645:
2643:
2636:
2631:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2569:cuboctahedron
2567:
2565:
2562:
2561:
2559:
2554:
2550:
2544:
2539:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2509:
2507:
2503:
2498:
2494:
2490:
2482:
2477:
2475:
2470:
2468:
2463:
2462:
2459:
2453:
2450:
2448:
2445:
2443:
2440:
2435:
2434:
2429:
2426:
2421:
2416:
2415:
2410:
2407:
2402:
2399:
2396:
2395:
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2347:
2332:
2326:
2319:
2309:Nahin, Paul.
2306:
2299:. p. 67.
2298:
2291:
2283:
2281:9780195341447
2277:
2273:
2269:
2268:
2260:
2252:
2251:
2243:
2239:
2225:
2221:
2220:
2215:
2211:
2210:
2209:
2203:
2199:
2193:
2189:
2179:
2176:
2175:
2166:
2163:
2161:
2158:
2155:
2149:
2145:
2141:
2138:
2134:
2131:
2127:
2124:
2121:
2120:
2114:
2113:
2108:
2107:Stanislaw Lem
2104:
2100:
2097:
2096:field of view
2093:
2089:
2085:
2082:
2078:
2075:
2071:
2067:
2063:
2060:
2057:
2054:
2050:
2049:step pyramids
2046:
2043:
2040:
2036:
2032:
2028:
2027:
2023:
2019:
2010:
2005:
1998:
1979:
1975:
1969:
1964:
1960:
1956:
1951:
1946:
1942:
1938:
1935:
1931:
1925:
1921:
1917:
1912:
1908:
1903:
1898:
1894:
1885:
1868:
1865:
1861:
1855:
1851:
1847:
1842:
1838:
1833:
1829:
1820:
1803:
1796:
1792:
1788:
1783:
1778:
1772:
1768:
1764:
1759:
1755:
1750:
1743:
1740:
1731:
1717:
1710:
1700:
1691:
1676:
1672:
1665:
1656:
1651:
1635:
1630:
1627:
1622:
1619:
1615:
1609:
1604:
1600:
1596:
1591:
1587:
1581:
1577:
1573:
1568:
1563:
1559:
1554:
1548:
1544:
1541:
1535:
1532:
1525:
1524:
1523:
1522:
1514:
1513:
1509:
1501:
1492:
1487:
1471:
1467:
1461:
1456:
1452:
1448:
1443:
1439:
1433:
1429:
1425:
1420:
1415:
1411:
1406:
1400:
1396:
1393:
1387:
1384:
1377:
1376:
1375:
1374:
1370:
1369:
1368:
1365:
1363:
1359:
1341:
1337:
1331:
1327:
1323:
1316:
1312:
1306:
1302:
1296:
1291:
1287:
1282:
1276:
1273:
1268:
1265:
1258:
1257:
1256:
1239:
1234:
1228:
1224:
1220:
1213:
1209:
1203:
1199:
1193:
1188:
1184:
1173:
1172:
1171:
1170:is obtained:
1165:
1156:
1151:
1150:Heronian mean
1135:
1128:Distributing
1126:
1123:
1116:
1109:
1089:
1084:
1078:
1073:
1069:
1065:
1060:
1056:
1050:
1046:
1042:
1037:
1032:
1028:
1021:
1013:
1009:
1005:
1000:
996:
989:
986:
979:
978:
977:
973:
969:
965:
961:
957:
953:
949:
929:
924:
918:
913:
909:
905:
900:
895:
891:
884:
881:
876:
870:
865:
861:
857:
852:
848:
844:
839:
834:
830:
826:
821:
817:
810:
807:
800:
799:
798:
792:
783:
778:
775:, and of the
762:
739:
736:
733:
725:
721:
715:
711:
703:
699:
693:
689:
682:
675:
670:
666:
660:
656:
650:
643:
638:
634:
628:
624:
614:
613:
612:
609:
603:
594:
585:
576:
556:
551:
545:
541:
535:
531:
527:
522:
518:
512:
508:
501:
498:
491:
490:
489:
487:
482:
479:
450:
446:
440:
436:
432:
429:
426:
423:
418:
414:
409:
403:
400:
395:
392:
385:
384:
383:
374:
370:
366:
351:
349:
345:
340:
338:
334:
329:
326:
320:
315:
307:
298:
296:
295:
288:
286:
285:
280:
279:
274:
273:
268:
264:
263:right pyramid
260:
259:
258:right frustum
254:
250:
246:
242:
238:
234:
230:
226:
218:
212:
206:
194:
188:
184:
179:
176:
172:
168:
164:
161:
157:
155:
151:
148:
145:
139:
132:
130:
126:
122:
117:
115:
111:
107:
102:
100:
96:
93:
82:
75:
73:
69:
62:
55:
46:
41:
36:
27:
22:
2860:
2823:
2779:trapezohedra
2730:
2723:
2527:dodecahedron
2431:
2412:
2356:
2352:
2346:
2334:. Retrieved
2325:
2310:
2305:
2296:
2290:
2266:
2259:
2249:
2242:
2217:
2213:
2205:
2197:
2192:
2128:and typical
2116:
2112:The Cyberiad
2110:
2003:
1996:
1994:
1883:
1818:
1709:slant height
1706:
1685:Surface area
1663:
1654:
1499:
1490:
1366:
1356:
1254:
1163:
1154:
1127:
1121:
1114:
1107:
1104:
976:, one gets:
971:
967:
963:
959:
955:
951:
947:
944:
790:
781:
754:
610:
601:
592:
583:
574:
571:
483:
480:
465:
373:13th dynasty
362:
341:
330:
327:
324:
292:
282:
276:
271:
270:
257:
256:
235:(normally a
228:
224:
210:
202:
192:
143:
137:
120:
105:
60:
53:
2549:semiregular
2532:icosahedron
2512:tetrahedron
2115:, the poem
337:prismatoids
253:trapezoidal
158:asymmetric
2875:Categories
2844:prismatoid
2774:bipyramids
2758:antiprisms
2732:hosohedron
2522:octahedron
2234:References
2130:lampshades
319:octahedron
317:A regular
166:Properties
81:trapezoids
79:isosceles
2881:Polyhedra
2839:birotunda
2829:bifrustum
2594:snub cube
2489:polyhedra
2433:MathWorld
2414:MathWorld
2381:125509773
2196:The term
2117:Love and
2045:Ziggurats
1895:π
1830:π
1765:−
1628:π
1623:
1519:-gons is:
1394:π
1152:of areas
1136:α
1022:α
1006:−
906:−
885:α
858:α
845:−
827:α
763:α
737:α
528:−
348:bifrustum
344:congruent
267:truncated
249:polygonal
160:bipyramid
2819:bicupola
2799:pyramids
2725:dihedron
2214:frustrum
2172:See also
2074:Illinois
2013:Examples
354:Formulas
275:. In a
229:frustums
205:geometry
114:Vertices
88:regular
2861:italics
2849:scutoid
2834:rotunda
2824:frustum
2553:uniform
2502:regular
2487:Convex
2361:Bibcode
2336:17 July
2313:√
2224:Plautus
2208:frustum
2198:frustum
2167:candies
2126:Buckets
2103:English
2101:In the
2070:Chicago
237:pyramid
211:frustum
2814:cupola
2767:duals:
2753:prisms
2379:
2278:
1995:where
1652:where
1488:where
1105:where
797:only:
572:where
486:volume
466:where
359:Volume
225:frusta
169:convex
142:, , (*
2377:S2CID
2204:
2202:Latin
2184:Notes
1508:radii
294:prism
261:is a
239:or a
233:solid
217:Latin
99:Edges
92:-gons
72:Faces
2517:cube
2338:2011
2276:ISBN
2165:Rolo
2086:The
2079:The
2064:The
2022:Rolo
2002:and
1661:and
1497:and
1161:and
788:and
599:and
581:and
484:The
470:and
333:apex
255:. A
241:cone
207:, a
58:and
2551:or
2369:doi
2090:in
2068:in
1730:is
1620:cot
954:= (
382:):
289:not
281:or
227:or
221:pl.
203:In
195:= 3
175:Net
63:= 4
56:= 5
2877::
2430:.
2411:.
2375:.
2367:.
2357:72
2355:.
2315:−1
2270:.
2072:,
2047:,
1549:12
1120:=
1113:−
970:+
968:ab
966:+
962:)(
958:−
950:−
377:c.
350:.
339:.
223::
144:nn
83:,
2863:.
2555:)
2547:(
2504:)
2500:(
2480:e
2473:t
2466:v
2436:.
2417:.
2383:.
2371::
2363::
2340:.
2318:.
2284:.
2061:.
2041:.
2007:2
2004:r
2000:1
1997:r
1980:,
1976:)
1970:2
1965:2
1961:r
1957:+
1952:2
1947:1
1943:r
1939:+
1936:s
1932:)
1926:2
1922:r
1918:+
1913:1
1909:r
1904:(
1899:(
1869:,
1866:s
1862:)
1856:2
1852:r
1848:+
1843:1
1839:r
1834:(
1804:,
1797:2
1793:h
1789:+
1784:2
1779:)
1773:2
1769:r
1760:1
1756:r
1751:(
1744:=
1741:s
1718:s
1667:2
1664:a
1658:1
1655:a
1636:,
1631:n
1616:)
1610:2
1605:2
1601:a
1597:+
1592:2
1588:a
1582:1
1578:a
1574:+
1569:2
1564:1
1560:a
1555:(
1545:h
1542:n
1536:=
1533:V
1517:n
1510:.
1503:2
1500:r
1494:1
1491:r
1472:,
1468:)
1462:2
1457:2
1453:r
1449:+
1444:2
1440:r
1434:1
1430:r
1426:+
1421:2
1416:1
1412:r
1407:(
1401:3
1397:h
1388:=
1385:V
1342:.
1338:)
1332:2
1328:B
1324:+
1317:2
1313:B
1307:1
1303:B
1297:+
1292:1
1288:B
1283:(
1277:3
1274:h
1269:=
1266:V
1240:;
1235:3
1229:2
1225:B
1221:+
1214:2
1210:B
1204:1
1200:B
1194:+
1189:1
1185:B
1167:2
1164:B
1158:1
1155:B
1122:h
1118:2
1115:h
1111:1
1108:h
1090:,
1085:3
1079:2
1074:2
1070:h
1066:+
1061:2
1057:h
1051:1
1047:h
1043:+
1038:2
1033:1
1029:h
1019:)
1014:2
1010:h
1001:1
997:h
993:(
990:=
987:V
974:)
972:b
964:a
960:b
956:a
952:b
948:a
930:.
925:3
919:3
914:2
910:h
901:3
896:1
892:h
882:=
877:3
871:2
866:2
862:h
853:2
849:h
840:2
835:1
831:h
822:1
818:h
811:=
808:V
794:2
791:h
785:1
782:h
740:,
734:=
726:2
722:h
716:1
712:h
704:2
700:B
694:1
690:B
683:=
676:2
671:2
667:h
661:2
657:B
651:=
644:2
639:1
635:h
629:1
625:B
605:2
602:h
596:1
593:h
587:2
584:B
578:1
575:B
557:,
552:3
546:2
542:B
536:2
532:h
523:1
519:B
513:1
509:h
502:=
499:V
476:h
472:b
468:a
451:,
447:)
441:2
437:b
433:+
430:b
427:a
424:+
419:2
415:a
410:(
404:3
401:h
396:=
393:V
375:(
215:(
197:)
193:n
146:)
140:v
138:n
135:C
121:n
119:2
106:n
104:3
90:n
85:2
77:n
65:)
61:n
54:n
51:(
31:n
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
