506:
1079:
354:
1669:
2611:
2600:
852:
49:
1013:. However, the contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) of
474:
Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. Though highly popularized, Archimedes probably does not submerge
893:
Commonly used prefixes for cubed length units are the cubic millimetre (mm), cubic centimetre (cm), cubic decimetre (dm), cubic metre (m) and the cubic kilometre (km). The conversion between the prefix units are as follows: 1000 mm = 1 cm, 1000 cm = 1 dm, and 1000 dm =
890:. When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm = 2.3 (cm) = 2.3 (0.01 m) = 0.0000023 m (five zeros).
902:(L) as a unit of volume, where 1 L = 1 dm = 1000 cm = 0.001 m. For the litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L.
557:
as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz).
656:
atoms unbounded the metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the
International Prototype Metre. The definition of the metre was redefined again in 1983 to use the
2473:
1659:
369:
The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz). The earliest evidence of volume calculation came from
775:
or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure
1203:
711:(order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to
402:, ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. The Egyptians use their units of length (the
1356:
1539:
414:) to devise their units of volume, such as the volume cubit or deny (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit).
1475:
2460:
2105:
2318:
1421:
1585:
1565:
826:(160,000,000 L) of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made.
505:
1594:
475:
the golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. Instead, he likely have devised a primitive form of a
2361:
1811:
569:, which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by
1261:
1232:
767:
improved, small volumes nowadays are usually measured using standardized human-made containers. This method is common for measuring small volume of fluids or
2264:
455:
approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by
2169:
A History of the Metric System of
Weights and Measures, with Emphasis on Campaigns for its Adoption in Great Britain, and in The United States Prior to 1914
1109:
294:
and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of
1035:
2017:
2514:
2441:
2401:
2269:
2236:
2143:
1976:
1912:
743:. However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable
1270:
2092:
309:
In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple
177:
166:
1490:
2579:
2468:
2331:
2203:
629:, for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824, the
545:. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the
2135:
1968:
611:
On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the
1426:
2068:
2541:
2498:
2366:
1816:
1480:
910:
645:
638:
582:
275:
2100:
846:
606:
1700:
1588:
1544:
395:
1106:
method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:
2393:
1904:
298:(gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By
1751:
1863:
1838:
1267:
method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as:
590:
218:
1734:
783:
597:
in the 17th and 18th centuries to form the modern integral calculus, which remains in use in the 21st century.
712:
566:
484:
1082:
Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the volume of rotated f(x).
739:
The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and
2195:
1366:
696:
670:
538:
500:
255:
31:
1374:
772:
716:
337:
325:
2536:
2326:
2172:
1727:
692:
562:
329:
247:
2428:
2052:
1723:
1359:
876:
594:
565:
applied the philosophy of modern integral calculus to calculate the volume of any object. He devised
547:
542:
452:
418:
251:
230:
2603:
2009:
637:
of water at 17 °C (62 °F). This definition was further refined until the United
Kingdom's
353:
1091:
496:
476:
2252:
719:
of three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus is the
537:. The sheer quantity of such units motivated British kings to standardize them, culminated in the
526:
1756:
807:
777:
708:
188:
1379:
2564:
802:
are adequate for cooking and daily life applications, however, they are not precise enough for
2614:
2575:
2510:
2437:
2407:
2397:
2232:
2209:
2199:
2149:
2139:
2013:
1990:
1982:
1972:
1918:
1908:
1570:
1264:
768:
724:
321:
310:
1550:
641:, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water.
2610:
2599:
2060:
1103:
1053:
1009:
Capacity is the maximum amount of material that a container can hold, measured in volume or
815:
764:
666:
570:
207:
479:. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a
2630:
1716:
1484:
1370:
1099:
1073:
1057:
872:
851:
799:
756:
630:
399:
333:
303:
259:
2056:
1078:
2503:
906:
842:
740:
720:
682:
658:
480:
425:
424:, written in around 300 BCE, detailed the exact formulas for calculating the volume of
271:
1668:
1237:
1208:
2624:
1738:
979:
967:
895:
883:
823:
795:
649:
371:
340:
and higher dimensions, an analogous concept to the normal volume is the hypervolume.
144:
54:
48:
2040:
1761:
1680:
939:
819:
791:
634:
586:
578:
530:
436:. The formula were determined by prior mathematicians by using a primitive form of
411:
152:
863:(with a side length of one). Because the volume occupies three dimensions, if the
302:, the term "volume" sometimes is used to refer to the corresponding region (e.g.,
1780:
At constant temperature and pressure, ignoring other states of matter for brevity
2571:
2176:
1688:
1095:
985:
947:
943:
887:
868:
833:, the container's volume is modeled by shapes and calculated using mathematics.
686:
574:
534:
518:
464:
460:
407:
375:
263:
132:
112:
96:
66:
2531:
2356:
1806:
1654:{\displaystyle \iiint _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi .}
1994:
1052:, they have an essentially the same volume calculation formula as one for the
961:
933:
925:
921:
917:
803:
760:
653:
510:
441:
314:
287:
140:
136:
70:
62:
2411:
2153:
1922:
2213:
929:
860:
859:
To ease calculations, a unit of volume is equal to the volume occupied by a
855:
Some SI units of volume to scale and approximate corresponding mass of water
830:
744:
867:(m) is chosen as a unit of length, the corresponding unit of volume is the
440:, by breaking the shapes into smaller and simpler pieces. A century later,
1864:"IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain""
1730:
is the volume of fluid which passes through a given surface per unit time.
1017:
will not be able to contain the same 7,200 t (15,900,000 lb) of
365:, an ancient municipal institution for the control of weights and measures
1672:
1087:
1049:
1014:
997:
957:
728:
613:
437:
383:
299:
128:
2436:(9th ed.). International Bureau of Weights and Measures. Dec 2022.
2294:
378:
as mathematical problems, approximating volume of simple shapes such as
1706:
1684:
1018:
811:
787:
752:
699:, volume cannot be physically measured as a negative value, similar to
553:
456:
429:
387:
362:
317:
91:
17:
2064:
509:
Diagram showing how to measure volume using a graduated cylinder with
1198:{\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx}
1061:
1045:
1010:
993:
975:
700:
662:
522:
433:
379:
291:
279:
199:
116:
58:
1986:
1077:
971:
899:
864:
850:
748:
619:
504:
468:
403:
352:
295:
283:
267:
195:
159:
120:
108:
794:
to measure volume of fluids at the microscopic scale. Calibrated
451:) devised approximate volume formula of several shapes using the
1710:
1041:
989:
951:
704:
625:
391:
124:
203:
2268:. Consultative Committee for Length. 20 May 2019. p. 1.
2132:
Scientific Unit
Conversion: A Practical Guide to Metrication
483:
submerged underwater, which will tip accordingly due to the
320:. Volumes of more complicated shapes can be calculated with
1719:
is total volume divided by mass, or the inverse of density.
818:. The largest of such calibrated containers are petroleum
1021:, due to naphtha's lower density and thus larger volume.
723:; this formulation is useful when working with different
521:, many units for measuring volume were made, such as the
1839:"IEC 60050 — Details for IEV number 102-04-40: "volume""
551:(medicine compound catalog) adopted the Roman gallon or
1713:
per unit volume, or total mass divided by total volume.
1240:
1211:
1965:
1691:
explicitly define its volume and surface properties.
1597:
1573:
1553:
1493:
1429:
1382:
1273:
1112:
313:
shapes can have their volume easily calculated using
1351:{\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx}
1090:
calculus. One of which is calculating the volume of
2171:(PhD thesis). Indiana University. pp. 99–100.
1683:is a representation of the object's surface, using
229:
213:
187:
176:
165:
150:
102:
90:
78:
41:
2563:
2502:
1901:Mathematics in Ancient Egypt: A Contextual History
1653:
1579:
1559:
1533:
1469:
1415:
1350:
1255:
1226:
1197:
623: (1 dm) for volumes of liquid; and the
1894:
1892:
1890:
1888:
1886:
1884:
1534:{\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,}
394:. These math problems have been written in the
2509:(6th ed.). Brooks Cole Cengage Learning.
2362:National Institute of Standards and Technology
2041:"Just what did Archimedes say about buoyancy?"
1958:
1956:
1954:
1952:
1812:National Institute of Standards and Technology
324:if a formula exists for the shape's boundary.
1950:
1948:
1946:
1944:
1942:
1940:
1938:
1936:
1934:
1932:
1086:The calculation of volume is a vital part of
806:. There, volume of liquids is measured using
633:was defined to be the volume occupied by ten
617: (1 m) for volume of firewood; the
8:
2562:Cengel, Yunus A.; Boles, Michael A. (2002).
2423:
2421:
2265:International Bureau of Weights and Measures
913:units of volume are also in use, including:
644:The 1960 redefinition of the metre from the
2455:
2453:
1801:
1799:
1797:
1423:over the region. It is usually written as:
258:. It is often quantified numerically using
2257:for the definition of the metre in the SI"
2087:
2085:
1587:measured from the polar axis; see more on
47:
2390:A Student's Guide to Dimensional Analysis
1868:International Electrotechnical Vocabulary
1843:International Electrotechnical Vocabulary
1641:
1634:
1627:
1612:
1602:
1596:
1572:
1552:
1521:
1514:
1507:
1498:
1492:
1470:{\displaystyle \iiint _{D}1\,dx\,dy\,dz.}
1457:
1450:
1443:
1434:
1428:
1381:
1341:
1336:
1304:
1295:
1290:
1272:
1239:
1210:
1188:
1177:
1155:
1131:
1126:
1111:
1667:
2566:Thermodynamics: an engineering approach
1793:
1773:
1741:of the substance divided by its volume.
1036:List of formulas in elementary geometry
61:. This cup measures volume in units of
2295:"Volume - Encyclopedia of Mathematics"
1547:(using the convention for angles with
38:
2231:. CBS College Publishing. p. 3.
829:For even larger volumes such as in a
822:, some can hold up to 1,000,000
491:Calculus and standardization of units
7:
2479:from the original on 2 February 2022
1263:are the plane curve boundaries. The
357:6 volumetric measures from the
2369:from the original on August 7, 2022
2275:from the original on 13 August 2022
2229:Physics For Science and Engineering
2130:Cardarelli, François (6 Dec 2012).
1819:from the original on August 7, 2022
27:Quantity of three-dimensional space
2020:from the original on 11 March 2009
57:can be used to measure volumes of
25:
2469:Rochester Institute of Technology
2430:The International System of Units
1102:on the same plane. The washer or
697:Euclidean three-dimensional space
2609:
2598:
2388:Lemons, Don S. (16 March 2017).
2111:from the original on 20 May 2022
2093:"Balances, Weights and Measures"
882:The metric units of volume uses
2544:from the original on 2022-08-09
2505:Calculus: Early Transcendentals
2136:Springer Science+Business Media
2071:from the original on 2021-04-14
1969:Springer Science+Business Media
1060:of the shape multiplied by its
871:(m). The cubic metre is also a
561:Around the early 17th century,
1404:
1386:
1337:
1333:
1327:
1318:
1312:
1305:
1250:
1244:
1221:
1215:
1174:
1167:
1152:
1145:
1:
2167:Cox, Edward Franklin (1958).
646:International Prototype Metre
639:Weights and Measures Act 1985
601:Metrication and redefinitions
445:
2101:Royal Pharmaceutical Society
1040:For many shapes such as the
847:Orders of magnitude (volume)
671:reworded for clarity in 2019
607:History of the metric system
1701:List of physical quantities
665:(which is derived from the
463:in the 5th century CE, the
396:Moscow Mathematical Papyrus
336:objects have no volume; in
2647:
2604:Perimeters, Areas, Volumes
2530:Benson, Tom (7 May 2021).
2394:Cambridge University Press
1963:Treese, Steven A. (2018).
1905:Princeton University Press
1899:Imhausen, Annette (2016).
1698:
1675:triangle mesh of a dolphin
1591:), the volume integral is
1416:{\displaystyle f(x,y,z)=1}
1369:is given by the triple or
1071:
1033:
875:. Therefore, volume has a
840:
680:
604:
494:
29:
2227:Marion, Jerry B. (1982).
2104:. 4 Feb 2020. p. 1.
747:found in nature, such as
591:Gottfried Wilhelm Leibniz
46:
2461:"Volumes by Integration"
2134:(2nd ed.). London:
1735:volumetric heat capacity
1580:{\displaystyle \varphi }
784:Air displacement pipette
417:The last three books of
2196:Oxford University Press
2190:Cook, James L. (1991).
1560:{\displaystyle \theta }
1481:cylindrical coordinates
1367:three-dimensional space
539:Assize of Bread and Ale
459:in the 3rd century CE,
256:three-dimensional space
32:Volume (disambiguation)
2319:"Use of Micropipettes"
2299:encyclopediaofmath.org
1676:
1655:
1581:
1561:
1535:
1471:
1417:
1352:
1257:
1228:
1199:
1083:
856:
717:infinitesimal calculus
707:. Like all continuous
514:
398:(c. 1820 BCE). In the
366:
2537:Glenn Research Center
2472:. 22 September 2014.
2365:. February 25, 2022.
2327:Buffalo State College
1967:. Cham, Switzerland:
1752:Banach–Tarski paradox
1671:
1656:
1582:
1562:
1545:spherical coordinates
1536:
1472:
1418:
1353:
1258:
1229:
1200:
1081:
1072:Further information:
854:
713:Cavalieri's principle
681:Further information:
605:Further information:
567:Cavalieri's principle
563:Bonaventura Cavalieri
508:
495:Further information:
485:Archimedes' principle
356:
290:). The definition of
2039:Graf, E. H. (2004).
1724:volumetric flow rate
1595:
1571:
1551:
1491:
1427:
1380:
1271:
1238:
1209:
1110:
1092:solids of revolution
595:Maria Gaetana Agnesi
548:London Pharmacopoeia
543:Henry III of England
501:Apothecaries' system
453:method of exhaustion
449: 287 – 212 BCE
219:coord transformation
30:For other uses, see
2057:2004PhTea..42..296G
2045:The Physics Teacher
1807:"SI Units - Volume"
1709:is the substance's
1567:as the azimuth and
1300:
1136:
1005:Capacity and volume
808:graduated cylinders
778:cooking ingredients
541:statute in 1258 by
497:History of calculus
477:hydrostatic balance
2192:Conversion Factors
2010:"The Golden Crown"
1815:. April 13, 2022.
1757:Dimensional weight
1695:Derived quantities
1677:
1664:Geometric modeling
1651:
1577:
1557:
1531:
1467:
1413:
1348:
1286:
1253:
1224:
1195:
1122:
1084:
898:also includes the
857:
769:granular materials
725:coordinate systems
648:to the orange-red
515:
367:
276:US customary units
154:SI base units
2516:978-0-495-01166-8
2443:978-92-822-2272-0
2403:978-1-107-16115-3
2357:"Area and Volume"
2238:978-4-8337-0098-6
2145:978-1-4471-0805-4
2065:10.1119/1.1737965
2014:Drexel University
1978:978-3-319-77577-7
1914:978-1-4008-7430-9
1265:shell integration
1256:{\textstyle g(x)}
1227:{\textstyle f(x)}
1068:Integral calculus
816:volumetric flasks
432:, cylinders, and
322:integral calculus
311:three-dimensional
241:
240:
16:(Redirected from
2638:
2613:
2602:
2586:
2585:
2569:
2559:
2553:
2552:
2550:
2549:
2527:
2521:
2520:
2508:
2495:
2489:
2488:
2486:
2484:
2478:
2465:
2457:
2448:
2447:
2435:
2425:
2416:
2415:
2385:
2379:
2378:
2376:
2374:
2353:
2347:
2346:
2344:
2342:
2337:on 4 August 2016
2336:
2330:. Archived from
2323:
2315:
2309:
2308:
2306:
2305:
2291:
2285:
2284:
2282:
2280:
2274:
2261:
2255:Mise en pratique
2249:
2243:
2242:
2224:
2218:
2217:
2198:. pp. xvi.
2187:
2181:
2180:
2164:
2158:
2157:
2127:
2121:
2120:
2118:
2116:
2110:
2097:
2089:
2080:
2079:
2077:
2076:
2036:
2030:
2029:
2027:
2025:
2005:
1999:
1998:
1960:
1927:
1926:
1896:
1879:
1878:
1876:
1875:
1860:
1854:
1853:
1851:
1850:
1835:
1829:
1828:
1826:
1824:
1803:
1781:
1778:
1660:
1658:
1657:
1652:
1617:
1616:
1607:
1606:
1586:
1584:
1583:
1578:
1566:
1564:
1563:
1558:
1540:
1538:
1537:
1532:
1503:
1502:
1476:
1474:
1473:
1468:
1439:
1438:
1422:
1420:
1419:
1414:
1373:of the constant
1358:The volume of a
1357:
1355:
1354:
1349:
1340:
1308:
1299:
1294:
1262:
1260:
1259:
1254:
1233:
1231:
1230:
1225:
1204:
1202:
1201:
1196:
1187:
1183:
1182:
1181:
1160:
1159:
1135:
1130:
1104:disc integration
1094:, by rotating a
765:glass production
667:caesium standard
571:Pierre de Fermat
450:
447:
270:) or by various
260:SI derived units
221:
155:
51:
39:
21:
2646:
2645:
2641:
2640:
2639:
2637:
2636:
2635:
2621:
2620:
2595:
2590:
2589:
2582:
2561:
2560:
2556:
2547:
2545:
2529:
2528:
2524:
2517:
2497:
2496:
2492:
2482:
2480:
2476:
2463:
2459:
2458:
2451:
2444:
2433:
2427:
2426:
2419:
2404:
2387:
2386:
2382:
2372:
2370:
2355:
2354:
2350:
2340:
2338:
2334:
2321:
2317:
2316:
2312:
2303:
2301:
2293:
2292:
2288:
2278:
2276:
2272:
2259:
2251:
2250:
2246:
2239:
2226:
2225:
2221:
2206:
2189:
2188:
2184:
2166:
2165:
2161:
2146:
2138:. p. 151.
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2108:
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2090:
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2037:
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2021:
2008:Rorres, Chris.
2007:
2006:
2002:
1979:
1962:
1961:
1930:
1915:
1898:
1897:
1882:
1873:
1871:
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1820:
1805:
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1795:
1790:
1785:
1784:
1779:
1775:
1770:
1748:
1717:Specific volume
1703:
1697:
1666:
1608:
1598:
1593:
1592:
1569:
1568:
1549:
1548:
1494:
1489:
1488:
1485:volume integral
1430:
1425:
1424:
1378:
1377:
1371:volume integral
1269:
1268:
1236:
1235:
1207:
1206:
1173:
1151:
1141:
1137:
1108:
1107:
1076:
1074:Volume integral
1070:
1038:
1032:
1027:
1007:
884:metric prefixes
873:SI derived unit
849:
841:Main articles:
839:
759:. Later on, as
751:, sheep or pig
737:
689:
679:
631:imperial gallon
609:
603:
503:
493:
448:
426:parallelepipeds
400:Reisner Papyrus
351:
349:Ancient history
346:
334:two-dimensional
304:bounding volume
222:
217:
216:
215:Behaviour under
153:
105:
81:
74:
35:
28:
23:
22:
15:
12:
11:
5:
2644:
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2634:
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2623:
2622:
2619:
2618:
2607:
2594:
2593:External links
2591:
2588:
2587:
2580:
2574:. p. 11.
2554:
2522:
2515:
2499:Stewart, James
2490:
2449:
2442:
2417:
2402:
2396:. p. 38.
2380:
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2310:
2286:
2244:
2237:
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2204:
2182:
2159:
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2122:
2081:
2051:(5): 296–299.
2031:
2000:
1977:
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1001:
983:
965:
955:
937:
911:U.S. customary
905:Various other
894:1 m. The
886:, strictly in
877:unit dimension
843:Unit of volume
838:
835:
796:measuring cups
736:
733:
721:volume element
683:Volume element
678:
675:
659:speed of light
602:
599:
513:markings, 1926
492:
489:
481:weighing scale
350:
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80:Common symbols
79:
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52:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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2583:
2581:0-07-238332-1
2577:
2573:
2568:
2567:
2558:
2555:
2543:
2539:
2538:
2533:
2532:"Gas Density"
2526:
2523:
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2512:
2507:
2506:
2500:
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2019:
2015:
2011:
2004:
2001:
1996:
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1966:
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1870:(in Japanese)
1869:
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908:
903:
901:
897:
896:metric system
891:
889:
888:powers of ten
885:
880:
878:
874:
870:
866:
862:
853:
848:
844:
836:
834:
832:
827:
825:
821:
820:storage tanks
817:
813:
809:
805:
801:
797:
793:
789:
785:
781:
779:
774:
771:, by using a
770:
766:
762:
758:
754:
750:
746:
742:
734:
732:
730:
727:, spaces and
726:
722:
718:
714:
710:
706:
702:
698:
694:
688:
684:
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672:
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664:
660:
655:
651:
650:emission line
647:
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636:
632:
628:
627:
622:
621:
616:
615:
608:
600:
598:
596:
592:
588:
584:
583:James Gregory
580:
576:
572:
568:
564:
559:
556:
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427:
423:
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415:
413:
409:
405:
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397:
393:
389:
385:
381:
377:
373:
372:ancient Egypt
364:
360:
359:mens ponderia
355:
348:
343:
341:
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335:
331:
327:
323:
319:
316:
312:
307:
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293:
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285:
281:
278:(such as the
277:
273:
269:
265:
262:(such as the
261:
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245:
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107:
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98:
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83:
77:
72:
68:
64:
60:
56:
55:measuring cup
50:
45:
40:
37:
33:
19:
2617:at Wikibooks
2606:at Wikibooks
2565:
2557:
2546:. Retrieved
2535:
2525:
2504:
2493:
2481:. Retrieved
2467:
2429:
2392:. New York:
2389:
2383:
2371:. Retrieved
2360:
2351:
2339:. Retrieved
2332:the original
2325:
2313:
2302:. Retrieved
2298:
2289:
2277:. Retrieved
2263:
2254:
2247:
2228:
2222:
2191:
2185:
2168:
2162:
2131:
2125:
2113:. Retrieved
2099:
2073:. Retrieved
2048:
2044:
2034:
2022:. Retrieved
2003:
1964:
1900:
1872:. Retrieved
1867:
1858:
1847:. Retrieved
1842:
1833:
1821:. Retrieved
1810:
1776:
1762:Dimensioning
1681:polygon mesh
1678:
1542:
1478:
1362:
1085:
1039:
1030:Basic shapes
1008:
904:
892:
881:
858:
828:
804:laboratories
792:biochemistry
782:
738:
690:
643:
624:
618:
612:
610:
587:Isaac Newton
579:Isaac Barrow
560:
552:
546:
516:
473:
420:
416:
368:
358:
308:
243:
242:
235:
92:SI unit
84:
67:fluid ounces
36:
2572:McGraw-Hill
2194:. Oxford :
1689:volume mesh
1589:conventions
1096:plane curve
1025:Computation
948:fluid ounce
869:cubic metre
786:is used in
735:Measurement
715:and to the
687:Volume form
575:John Wallis
519:Middle Ages
465:Middle East
461:Zu Chongzhi
438:integration
376:Mesopotamia
264:cubic metre
113:fluid ounce
104:Other units
97:cubic metre
71:millilitres
2570:. Boston:
2548:2022-08-13
2304:2023-05-27
2075:2022-08-07
1995:1036766223
1987:2018940415
1874:2023-09-19
1849:2023-09-19
1788:References
1699:See also:
1034:See also:
962:tablespoon
934:cubic mile
926:cubic yard
922:cubic foot
918:cubic inch
761:metallurgy
745:containers
677:Properties
654:krypton-86
511:fluid dram
442:Archimedes
315:arithmetic
288:cubic inch
133:fluid dram
2483:12 August
2412:959922612
2373:August 7,
2279:13 August
2177:301905667
2154:828776235
2115:13 August
1923:934433864
1823:August 7,
1728:discharge
1646:φ
1639:θ
1632:ρ
1625:φ
1622:
1610:ρ
1600:∭
1575:φ
1555:θ
1519:θ
1496:∭
1432:∭
1322:−
1288:∫
1284:π
1162:−
1124:∫
1120:π
1098:around a
930:acre-foot
861:unit cube
831:reservoir
729:manifolds
709:monotonic
428:, cones,
419:Euclid's
384:cylinders
231:Dimension
225:conserved
202:, no for
189:Conserved
178:Intensive
167:Extensive
2625:Category
2542:Archived
2501:(2008).
2474:Archived
2367:Archived
2270:Archived
2214:22861139
2173:ProQuest
2106:Archived
2069:Archived
2024:24 March
2018:Archived
1817:Archived
1746:See also
1685:polygons
1673:Low poly
1375:function
1088:integral
1050:cylinder
1015:fuel oil
998:hogshead
958:teaspoon
907:imperial
812:pipettes
773:multiple
757:bladders
753:stomachs
430:pyramids
421:Elements
318:formulas
300:metonymy
272:imperial
194:yes for
2341:19 June
2053:Bibcode
1737:is the
1707:Density
1019:naphtha
788:biology
741:pinches
695:of the
693:measure
554:congius
517:In the
457:Liu Hui
434:spheres
388:frustum
380:cuboids
363:Pompeii
344:History
252:regions
248:measure
200:liquids
59:liquids
18:Volumes
2631:Volume
2615:Volume
2578:
2513:
2440:
2410:
2400:
2235:
2212:
2202:
2175:
2152:
2142:
1993:
1985:
1975:
1921:
1911:
1687:. The
1483:, the
1360:region
1205:where
1062:height
1056:: the
1046:cuboid
1011:weight
994:bushel
980:barrel
976:gallon
944:drachm
879:of L.
800:spoons
755:, and
749:gourds
701:length
669:) and
663:second
635:pounds
626:gramme
533:, and
523:sester
292:length
280:gallon
244:Volume
208:plasma
206:, and
196:solids
145:barrel
117:gallon
69:, and
42:Volume
2477:(PDF)
2464:(PDF)
2434:(PDF)
2335:(PDF)
2322:(PDF)
2273:(PDF)
2260:(PDF)
2109:(PDF)
2096:(PDF)
1768:Notes
1054:prism
972:quart
940:minim
900:litre
865:metre
837:Units
691:As a
620:litre
614:stère
531:coomb
527:amber
469:India
412:digit
404:cubit
392:cones
326:Zero-
296:fluid
284:quart
268:litre
246:is a
204:gases
121:quart
109:Litre
2576:ISBN
2511:ISBN
2485:2022
2438:ISBN
2408:OCLC
2398:ISBN
2375:2022
2343:2016
2281:2022
2233:ISBN
2210:OCLC
2200:ISBN
2150:OCLC
2140:ISBN
2117:2022
2026:2009
1991:OCLC
1983:LCCN
1973:ISBN
1919:OCLC
1909:ISBN
1825:2022
1733:The
1722:The
1711:mass
1234:and
1100:line
1058:base
1048:and
1042:cube
990:peck
986:cord
968:gill
952:pint
845:and
814:and
798:and
790:and
763:and
705:area
703:and
685:and
661:and
593:and
535:seam
499:and
467:and
408:palm
390:and
374:and
338:four
332:and
330:one-
266:and
198:and
125:pint
63:cups
2061:doi
1726:or
1619:sin
1543:In
1487:is
1479:In
1365:in
909:or
824:bbl
652:of
361:in
306:).
274:or
254:in
250:of
172:yes
151:In
129:tsp
2627::
2540:.
2534:.
2466:.
2452:^
2420:^
2406:.
2359:.
2324:.
2297:.
2262:.
2208:.
2148:.
2098:.
2084:^
2067:.
2059:.
2049:42
2047:.
2043:.
2016:.
2012:.
1989:.
1981:.
1971:.
1931:^
1917:.
1907:.
1903:.
1883:^
1866:.
1841:.
1809:.
1796:^
1679:A
1064:.
1044:,
996:,
992:,
988:,
978:,
974:,
970:,
960:,
950:,
946:,
942:,
932:,
928:,
924:,
920:,
810:,
780:.
731:.
673:.
589:,
585:,
581:,
577:,
573:,
529:,
525:,
487:.
471:.
446:c.
410:,
406:,
386:,
382:,
328:,
286:,
282:,
183:no
143:,
141:yd
139:,
137:in
135:,
131:,
127:,
123:,
119:,
115:,
111:,
65:,
53:A
2584:.
2551:.
2519:.
2487:.
2446:.
2414:.
2377:.
2345:.
2307:.
2283:.
2253:"
2241:.
2216:.
2179:.
2156:.
2119:.
2078:.
2063::
2055::
2028:.
1997:.
1925:.
1877:.
1852:.
1827:.
1649:.
1643:d
1636:d
1629:d
1614:2
1604:D
1529:,
1526:z
1523:d
1516:d
1512:r
1509:d
1505:r
1500:D
1465:.
1462:z
1459:d
1455:y
1452:d
1448:x
1445:d
1441:1
1436:D
1411:1
1408:=
1405:)
1402:z
1399:,
1396:y
1393:,
1390:x
1387:(
1384:f
1363:D
1346:x
1343:d
1338:|
1334:)
1331:x
1328:(
1325:g
1319:)
1316:x
1313:(
1310:f
1306:|
1302:x
1297:b
1292:a
1281:2
1278:=
1275:V
1251:)
1248:x
1245:(
1242:g
1222:)
1219:x
1216:(
1213:f
1193:x
1190:d
1185:|
1179:2
1175:)
1171:x
1168:(
1165:g
1157:2
1153:)
1149:x
1146:(
1143:f
1139:|
1133:b
1128:a
1117:=
1114:V
1000:.
982:;
964:;
954:;
936:;
444:(
236:L
191:?
180:?
169:?
160:m
85:V
73:.
34:.
20:)
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