495:
343:
1068:
1658:
2600:
2589:
841:
38:
1002:. 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
463:
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
882:
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 =
879:. 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).
891:(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.
546:
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).
645:
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
2462:
1648:
358:
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
764:
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
1192:
700:(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
391:, 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
1345:
1528:
403:) 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).
1464:
2449:
2094:
2307:
1410:
1574:
1554:
815:(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.
1583:
464:
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
558:, 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
1250:
1221:
756:
improved, small volumes nowadays are usually measured using standardized human-made containers. This method is common for measuring small volume of fluids or
444:
approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by
2158:
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
1098:
283:
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
2350:
2006:
1800:
515:
2258:
732:. However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable
2253:
1259:
2081:
298:
In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple
1479:
2320:
494:
618:, for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824, the
534:. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the
1024:
600:
On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the
1415:
2057:
2503:
2430:
2390:
2225:
2132:
1965:
1901:
2530:
2355:
1805:
166:
155:
2568:
2457:
2192:
1577:
1095:
method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:
2124:
1957:
287:(gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By
1852:
1827:
1256:
method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as:
586:
in the 17th and 18th centuries to form the modern integral calculus, which remains in use in the 21st century.
2487:
1469:
1071:
Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the volume of rotated f(x).
899:
728:
The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and
634:
627:
571:
264:
2089:
835:
595:
1689:
1533:
384:
2382:
1893:
219:
2417:
1740:
579:
554:
applied the philosophy of modern integral calculus to calculate the volume of any object. He devised
207:
2592:
1998:
626:
of water at 17 °C (62 °F). This definition was further refined until the United
Kingdom's
1723:
772:
701:
555:
473:
2241:
708:
of three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus is the
526:. The sheer quantity of such units motivated British kings to standardize them, culminated in the
2184:
1355:
685:
659:
527:
489:
244:
20:
1368:
2553:
1363:
791:
are adequate for cooking and daily life applications, however, they are not precise enough for
761:
705:
326:
314:
2603:
1559:
2525:
2315:
2161:
1716:
1539:
681:
630:, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water.
551:
318:
236:
998:
Capacity is the maximum amount of material that a container can hold, measured in volume or
2041:
1712:
1348:
865:
583:
536:
531:
468:. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a
441:
407:
240:
8:
1080:
485:
465:
2045:
812:
2492:
1745:
796:
766:
697:
413:, written in around 300 BCE, detailed the exact formulas for calculating the volume of
177:
1226:
1197:
2564:
2499:
2426:
2396:
2386:
2221:
2198:
2188:
2138:
2128:
2002:
1979:
1971:
1961:
1907:
1897:
1253:
757:
713:
329:
and higher dimensions, an analogous concept to the normal volume is the hypervolume.
310:
299:
2049:
2029:
1092:
1042:
804:
753:
655:
559:
425:. The formula were determined by prior mathematicians by using a primitive form of
342:
196:
852:(with a side length of one). Because the volume occupies three dimensions, if the
291:, the term "volume" sometimes is used to refer to the corresponding region (e.g.,
2619:
1769:
At constant temperature and pressure, ignoring other states of matter for brevity
1705:
1473:
1359:
1088:
1062:
1046:
861:
788:
745:
619:
388:
322:
292:
248:
2165:
822:, the container's volume is modeled by shapes and calculated using mathematics.
2520:
2345:
1795:
1643:{\displaystyle \iiint _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi .}
895:
831:
729:
709:
671:
647:
469:
414:
260:
1983:
1041:, they have an essentially the same volume calculation formula as one for the
2613:
2599:
2588:
2400:
2142:
1911:
1727:
968:
956:
884:
872:
784:
638:
360:
133:
43:
2202:
848:
To ease calculations, a unit of volume is equal to the volume occupied by a
844:
Some SI units of volume to scale and approximate corresponding mass of water
1750:
1669:
928:
856:(m) is chosen as a unit of length, the corresponding unit of volume is the
840:
808:
780:
623:
575:
567:
519:
429:, by breaking the shapes into smaller and simpler pieces. A century later,
400:
141:
1853:"IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain""
1719:
is the volume of fluid which passes through a given surface per unit time.
1067:
1006:
will not be able to contain the same 7,200 t (15,900,000 lb) of
354:, an ancient municipal institution for the control of weights and measures
2560:
1677:
1084:
974:
936:
932:
876:
857:
675:
563:
523:
507:
453:
449:
396:
364:
252:
121:
101:
85:
55:
2425:(9th ed.). International Bureau of Weights and Measures. Dec 2022.
2283:
1657:
367:
as mathematical problems, approximating volume of simple shapes such as
950:
922:
914:
910:
906:
792:
749:
688:, volume cannot be physically measured as a negative value, similar to
642:
499:
430:
303:
276:
129:
125:
59:
51:
37:
2053:
498:
Diagram showing how to measure volume using a graduated cylinder with
1187:{\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx}
918:
849:
819:
733:
1661:
1076:
1038:
1003:
986:
946:
717:
426:
372:
288:
117:
1975:
602:
1695:
1673:
1007:
800:
783:
to measure volume of fluids at the microscopic scale. Calibrated
776:
741:
542:
445:
440:) devised approximate volume formula of several shapes using the
418:
376:
351:
306:
80:
1050:
1034:
999:
982:
964:
689:
651:
511:
422:
368:
280:
268:
188:
105:
47:
2257:. Consultative Committee for Length. 20 May 2019. p. 1.
960:
888:
853:
737:
608:
457:
392:
284:
272:
256:
184:
148:
109:
97:
2121:
Scientific Unit
Conversion: A Practical Guide to Metrication
472:
submerged underwater, which will tip accordingly due to the
309:. Volumes of more complicated shapes can be calculated with
1708:
is total volume divided by mass, or the inverse of density.
1699:
1030:
978:
940:
693:
614:
380:
113:
807:. The largest of such calibrated containers are petroleum
1010:, due to naphtha's lower density and thus larger volume.
712:; this formulation is useful when working with different
510:, many units for measuring volume were made, such as the
192:
1828:"IEC 60050 — Details for IEV number 102-04-40: "volume""
540:(medicine compound catalog) adopted the Roman gallon or
479:
1702:
per unit volume, or total mass divided by total volume.
1229:
1200:
1954:
1680:
explicitly define its volume and surface properties.
1586:
1562:
1542:
1482:
1418:
1371:
1262:
1101:
302:
shapes can have their volume easily calculated using
1340:{\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx}
1079:
calculus. One of which is calculating the volume of
2160:(PhD thesis). Indiana University. pp. 99–100.
1672:is a representation of the object's surface, using
2552:
2491:
1890:Mathematics in Ancient Egypt: A Contextual History
1642:
1568:
1548:
1522:
1458:
1404:
1339:
1244:
1215:
1186:
612: (1 dm) for volumes of liquid; and the
2611:
589:
1883:
1881:
1879:
1877:
1875:
1873:
1523:{\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,}
383:. These math problems have been written in the
2498:(6th ed.). Brooks Cole Cengage Learning.
2351:National Institute of Standards and Technology
2030:"Just what did Archimedes say about buoyancy?"
1947:
1945:
1943:
1941:
1801:National Institute of Standards and Technology
313:if a formula exists for the shape's boundary.
1939:
1937:
1935:
1933:
1931:
1929:
1927:
1925:
1923:
1921:
1075:The calculation of volume is a vital part of
795:. There, volume of liquids is measured using
622:was defined to be the volume occupied by ten
606: (1 m) for volume of firewood; the
2551:Cengel, Yunus A.; Boles, Michael A. (2002).
2412:
2410:
2254:International Bureau of Weights and Measures
1870:
902:units of volume are also in use, including:
633:The 1960 redefinition of the metre from the
2550:
2444:
2442:
1790:
1788:
1786:
1412:over the region. It is usually written as:
247:. It is often quantified numerically using
2246:for the definition of the metre in the SI"
2118:
2076:
2074:
1918:
1576:measured from the polar axis; see more on
36:
2407:
2379:A Student's Guide to Dimensional Analysis
1857:International Electrotechnical Vocabulary
1832:International Electrotechnical Vocabulary
1630:
1623:
1616:
1510:
1503:
1496:
1459:{\displaystyle \iiint _{D}1\,dx\,dy\,dz.}
1446:
1439:
1432:
1330:
1177:
2439:
1887:
1783:
1656:
1066:
839:
493:
341:
2555:Thermodynamics: an engineering approach
2486:
2071:
1730:of the substance divided by its volume.
1025:List of formulas in elementary geometry
50:. This cup measures volume in units of
2612:
2518:
2376:
2284:"Volume - Encyclopedia of Mathematics"
2215:
1951:
1536:(using the convention for angles with
993:
2220:. CBS College Publishing. p. 3.
1683:
1652:
818:For even larger volumes such as in a
811:, some can hold up to 1,000,000
480:Calculus and standardization of units
2468:from the original on 2 February 2022
2178:
2027:
1252:are the plane curve boundaries. The
1056:
346:6 volumetric measures from the
2358:from the original on August 7, 2022
2264:from the original on 13 August 2022
2218:Physics For Science and Engineering
2155:
2119:Cardarelli, François (6 Dec 2012).
1808:from the original on August 7, 2022
16:Quantity of three-dimensional space
13:
2009:from the original on 11 March 2009
1996:
337:
46:can be used to measure volumes of
14:
2631:
2581:
2458:Rochester Institute of Technology
2419:The International System of Units
1091:on the same plane. The washer or
686:Euclidean three-dimensional space
2598:
2587:
2377:Lemons, Don S. (16 March 2017).
2100:from the original on 20 May 2022
2082:"Balances, Weights and Measures"
871:The metric units of volume uses
2544:
2533:from the original on 2022-08-09
2512:
2494:Calculus: Early Transcendentals
2480:
2370:
2338:
2300:
2276:
2234:
2209:
2172:
2149:
2125:Springer Science+Business Media
2112:
2060:from the original on 2021-04-14
1958:Springer Science+Business Media
1049:of the shape multiplied by its
1018:
860:(m). The cubic metre is also a
550:Around the early 17th century,
2021:
1990:
1845:
1820:
1763:
1393:
1375:
1326:
1322:
1316:
1307:
1301:
1294:
1239:
1233:
1210:
1204:
1163:
1156:
1141:
1134:
1013:
723:
1:
2156:Cox, Edward Franklin (1958).
1776:
665:
635:International Prototype Metre
628:Weights and Measures Act 1985
590:Metrication and redefinitions
434:
2090:Royal Pharmaceutical Society
1029:For many shapes such as the
836:Orders of magnitude (volume)
660:reworded for clarity in 2019
596:History of the metric system
7:
1734:
1690:List of physical quantities
654:(which is derived from the
452:in the 5th century CE, the
385:Moscow Mathematical Papyrus
325:objects have no volume; in
10:
2636:
2593:Perimeters, Areas, Volumes
2519:Benson, Tom (7 May 2021).
2383:Cambridge University Press
1952:Treese, Steven A. (2018).
1894:Princeton University Press
1888:Imhausen, Annette (2016).
1687:
1664:triangle mesh of a dolphin
1580:), the volume integral is
1405:{\displaystyle f(x,y,z)=1}
1358:is given by the triple or
1060:
1022:
864:. Therefore, volume has a
829:
669:
593:
483:
332:
18:
2216:Marion, Jerry B. (1982).
2093:. 4 Feb 2020. p. 1.
736:found in nature, such as
580:Gottfried Wilhelm Leibniz
218:
202:
176:
165:
154:
139:
91:
79:
67:
35:
30:
2450:"Volumes by Integration"
2123:(2nd ed.). London:
1756:
1724:volumetric heat capacity
1569:{\displaystyle \varphi }
825:
773:Air displacement pipette
406:The last three books of
2185:Oxford University Press
2179:Cook, James L. (1991).
1549:{\displaystyle \theta }
1470:cylindrical coordinates
1356:three-dimensional space
528:Assize of Bread and Ale
448:in the 3rd century CE,
245:three-dimensional space
21:Volume (disambiguation)
2308:"Use of Micropipettes"
2288:encyclopediaofmath.org
1665:
1644:
1570:
1550:
1524:
1460:
1406:
1341:
1246:
1217:
1188:
1072:
845:
706:infinitesimal calculus
696:. Like all continuous
503:
387:(c. 1820 BCE). In the
355:
2526:Glenn Research Center
2461:. 22 September 2014.
2354:. February 25, 2022.
2316:Buffalo State College
1956:. Cham, Switzerland:
1741:Banach–Tarski paradox
1660:
1645:
1571:
1551:
1534:spherical coordinates
1525:
1461:
1407:
1342:
1247:
1218:
1189:
1070:
1061:Further information:
843:
702:Cavalieri's principle
670:Further information:
594:Further information:
556:Cavalieri's principle
552:Bonaventura Cavalieri
497:
484:Further information:
474:Archimedes' principle
345:
279:). The definition of
2028:Graf, E. H. (2004).
1713:volumetric flow rate
1584:
1560:
1540:
1480:
1416:
1369:
1260:
1227:
1198:
1099:
1081:solids of revolution
584:Maria Gaetana Agnesi
537:London Pharmacopoeia
532:Henry III of England
490:Apothecaries' system
442:method of exhaustion
438: 287 – 212 BCE
208:coord transformation
19:For other uses, see
2046:2004PhTea..42..296G
2034:The Physics Teacher
1796:"SI Units - Volume"
1698:is the substance's
1556:as the azimuth and
1289:
1125:
994:Capacity and volume
797:graduated cylinders
767:cooking ingredients
530:statute in 1258 by
486:History of calculus
466:hydrostatic balance
2181:Conversion Factors
1999:"The Golden Crown"
1804:. April 13, 2022.
1746:Dimensional weight
1684:Derived quantities
1666:
1653:Geometric modeling
1640:
1566:
1546:
1520:
1456:
1402:
1337:
1275:
1242:
1213:
1184:
1111:
1073:
887:also includes the
846:
758:granular materials
714:coordinate systems
637:to the orange-red
504:
356:
265:US customary units
143:SI base units
2505:978-0-495-01166-8
2432:978-92-822-2272-0
2392:978-1-107-16115-3
2346:"Area and Volume"
2227:978-4-8337-0098-6
2134:978-1-4471-0805-4
2054:10.1119/1.1737965
2003:Drexel University
1967:978-3-319-77577-7
1903:978-1-4008-7430-9
1254:shell integration
1245:{\textstyle g(x)}
1216:{\textstyle f(x)}
1057:Integral calculus
805:volumetric flasks
421:, cylinders, and
311:integral calculus
300:three-dimensional
230:
229:
2627:
2602:
2591:
2575:
2574:
2558:
2548:
2542:
2541:
2539:
2538:
2516:
2510:
2509:
2497:
2484:
2478:
2477:
2475:
2473:
2467:
2454:
2446:
2437:
2436:
2424:
2414:
2405:
2404:
2374:
2368:
2367:
2365:
2363:
2342:
2336:
2335:
2333:
2331:
2326:on 4 August 2016
2325:
2319:. Archived from
2312:
2304:
2298:
2297:
2295:
2294:
2280:
2274:
2273:
2271:
2269:
2263:
2250:
2244:Mise en pratique
2238:
2232:
2231:
2213:
2207:
2206:
2187:. pp. xvi.
2176:
2170:
2169:
2153:
2147:
2146:
2116:
2110:
2109:
2107:
2105:
2099:
2086:
2078:
2069:
2068:
2066:
2065:
2025:
2019:
2018:
2016:
2014:
1994:
1988:
1987:
1949:
1916:
1915:
1885:
1868:
1867:
1865:
1864:
1849:
1843:
1842:
1840:
1839:
1824:
1818:
1817:
1815:
1813:
1792:
1770:
1767:
1649:
1647:
1646:
1641:
1606:
1605:
1596:
1595:
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1529:
1527:
1526:
1521:
1492:
1491:
1465:
1463:
1462:
1457:
1428:
1427:
1411:
1409:
1408:
1403:
1362:of the constant
1347:The volume of a
1346:
1344:
1343:
1338:
1329:
1297:
1288:
1283:
1251:
1249:
1248:
1243:
1222:
1220:
1219:
1214:
1193:
1191:
1190:
1185:
1176:
1172:
1171:
1170:
1149:
1148:
1124:
1119:
1093:disc integration
1083:, by rotating a
754:glass production
656:caesium standard
560:Pierre de Fermat
439:
436:
259:) or by various
249:SI derived units
210:
144:
40:
28:
27:
2635:
2634:
2630:
2629:
2628:
2626:
2625:
2624:
2610:
2609:
2584:
2579:
2578:
2571:
2549:
2545:
2536:
2534:
2517:
2513:
2506:
2485:
2481:
2471:
2469:
2465:
2452:
2448:
2447:
2440:
2433:
2422:
2416:
2415:
2408:
2393:
2375:
2371:
2361:
2359:
2344:
2343:
2339:
2329:
2327:
2323:
2310:
2306:
2305:
2301:
2292:
2290:
2282:
2281:
2277:
2267:
2265:
2261:
2248:
2240:
2239:
2235:
2228:
2214:
2210:
2195:
2177:
2173:
2154:
2150:
2135:
2127:. p. 151.
2117:
2113:
2103:
2101:
2097:
2084:
2080:
2079:
2072:
2063:
2061:
2026:
2022:
2012:
2010:
1997:Rorres, Chris.
1995:
1991:
1968:
1950:
1919:
1904:
1886:
1871:
1862:
1860:
1851:
1850:
1846:
1837:
1835:
1826:
1825:
1821:
1811:
1809:
1794:
1793:
1784:
1779:
1774:
1773:
1768:
1764:
1759:
1737:
1706:Specific volume
1692:
1686:
1655:
1601:
1597:
1591:
1587:
1585:
1582:
1581:
1561:
1558:
1557:
1541:
1538:
1537:
1487:
1483:
1481:
1478:
1477:
1474:volume integral
1423:
1419:
1417:
1414:
1413:
1370:
1367:
1366:
1360:volume integral
1325:
1293:
1284:
1279:
1261:
1258:
1257:
1228:
1225:
1224:
1199:
1196:
1195:
1166:
1162:
1144:
1140:
1130:
1126:
1120:
1115:
1100:
1097:
1096:
1065:
1063:Volume integral
1059:
1027:
1021:
1016:
996:
873:metric prefixes
862:SI derived unit
838:
830:Main articles:
828:
748:. Later on, as
740:, sheep or pig
726:
678:
668:
620:imperial gallon
598:
592:
492:
482:
437:
415:parallelepipeds
389:Reisner Papyrus
340:
338:Ancient history
335:
323:two-dimensional
293:bounding volume
211:
206:
205:
204:Behaviour under
142:
94:
70:
63:
24:
17:
12:
11:
5:
2633:
2623:
2622:
2608:
2607:
2596:
2583:
2582:External links
2580:
2577:
2576:
2569:
2563:. p. 11.
2543:
2511:
2504:
2488:Stewart, James
2479:
2438:
2431:
2406:
2391:
2385:. p. 38.
2369:
2337:
2299:
2275:
2233:
2226:
2208:
2193:
2171:
2148:
2133:
2111:
2070:
2040:(5): 296–299.
2020:
1989:
1966:
1917:
1902:
1869:
1844:
1819:
1781:
1780:
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1775:
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1123:
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1058:
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1020:
1017:
1015:
1012:
995:
992:
991:
990:
972:
954:
944:
926:
900:U.S. customary
894:Various other
883:1 m. The
875:, strictly in
866:unit dimension
832:Unit of volume
827:
824:
785:measuring cups
725:
722:
710:volume element
672:Volume element
667:
664:
648:speed of light
591:
588:
502:markings, 1926
481:
478:
470:weighing scale
339:
336:
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95:
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89:
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83:
77:
76:
71:
69:Common symbols
68:
65:
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41:
33:
32:
15:
9:
6:
4:
3:
2:
2632:
2621:
2618:
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2572:
2570:0-07-238332-1
2566:
2562:
2557:
2556:
2547:
2532:
2528:
2527:
2522:
2521:"Gas Density"
2515:
2507:
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2489:
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2256:
2255:
2247:
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2237:
2229:
2223:
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2200:
2196:
2194:0-19-856349-3
2190:
2186:
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2175:
2167:
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2159:
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2008:
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1993:
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1981:
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1891:
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1880:
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1876:
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1859:(in Japanese)
1858:
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1834:(in Japanese)
1833:
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1807:
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1725:
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905:
904:
903:
901:
897:
892:
890:
886:
885:metric system
880:
878:
877:powers of ten
874:
869:
867:
863:
859:
855:
851:
842:
837:
833:
823:
821:
816:
814:
810:
809:storage tanks
806:
802:
798:
794:
790:
786:
782:
778:
774:
770:
768:
763:
760:, by using a
759:
755:
751:
747:
743:
739:
735:
731:
721:
719:
716:, spaces and
715:
711:
707:
703:
699:
695:
691:
687:
683:
677:
673:
663:
661:
657:
653:
649:
644:
640:
639:emission line
636:
631:
629:
625:
621:
617:
616:
611:
610:
605:
604:
597:
587:
585:
581:
577:
573:
572:James Gregory
569:
565:
561:
557:
553:
548:
545:
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539:
538:
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529:
525:
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517:
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412:
411:
404:
402:
398:
394:
390:
386:
382:
378:
374:
370:
366:
362:
361:ancient Egypt
353:
349:
348:mens ponderia
344:
330:
328:
324:
320:
316:
312:
308:
305:
301:
296:
294:
290:
286:
282:
278:
274:
270:
267:(such as the
266:
262:
258:
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251:(such as the
250:
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107:
103:
99:
96:
90:
87:
84:
82:
78:
75:
72:
66:
61:
57:
53:
49:
45:
44:measuring cup
39:
34:
29:
26:
22:
2606:at Wikibooks
2595:at Wikibooks
2554:
2546:
2535:. Retrieved
2524:
2514:
2493:
2482:
2470:. Retrieved
2456:
2418:
2381:. New York:
2378:
2372:
2360:. Retrieved
2349:
2340:
2328:. Retrieved
2321:the original
2314:
2302:
2291:. Retrieved
2287:
2278:
2266:. Retrieved
2252:
2243:
2236:
2217:
2211:
2180:
2174:
2157:
2151:
2120:
2114:
2102:. Retrieved
2088:
2062:. Retrieved
2037:
2033:
2023:
2011:. Retrieved
1992:
1953:
1889:
1861:. Retrieved
1856:
1847:
1836:. Retrieved
1831:
1822:
1810:. Retrieved
1799:
1765:
1751:Dimensioning
1670:polygon mesh
1667:
1531:
1467:
1351:
1074:
1028:
1019:Basic shapes
997:
893:
881:
870:
847:
817:
793:laboratories
781:biochemistry
771:
727:
679:
632:
613:
607:
601:
599:
576:Isaac Newton
568:Isaac Barrow
549:
541:
535:
505:
462:
409:
405:
357:
347:
297:
232:
231:
224:
81:SI unit
73:
56:fluid ounces
25:
2561:McGraw-Hill
2183:. Oxford :
1678:volume mesh
1578:conventions
1085:plane curve
1014:Computation
937:fluid ounce
858:cubic metre
775:is used in
724:Measurement
704:and to the
676:Volume form
564:John Wallis
508:Middle Ages
454:Middle East
450:Zu Chongzhi
427:integration
365:Mesopotamia
253:cubic metre
102:fluid ounce
93:Other units
86:cubic metre
60:millilitres
2559:. Boston:
2537:2022-08-13
2293:2023-05-27
2064:2022-08-07
1984:1036766223
1976:2018940415
1863:2023-09-19
1838:2023-09-19
1777:References
1688:See also:
1023:See also:
951:tablespoon
923:cubic mile
915:cubic yard
911:cubic foot
907:cubic inch
750:metallurgy
734:containers
666:Properties
643:krypton-86
500:fluid dram
431:Archimedes
304:arithmetic
277:cubic inch
122:fluid dram
2472:12 August
2401:959922612
2362:August 7,
2268:13 August
2166:301905667
2143:828776235
2104:13 August
1912:934433864
1812:August 7,
1717:discharge
1635:φ
1628:θ
1621:ρ
1614:φ
1611:
1599:ρ
1589:∭
1564:φ
1544:θ
1508:θ
1485:∭
1421:∭
1311:−
1277:∫
1273:π
1151:−
1113:∫
1109:π
1087:around a
919:acre-foot
850:unit cube
820:reservoir
718:manifolds
698:monotonic
417:, cones,
408:Euclid's
373:cylinders
220:Dimension
214:conserved
191:, no for
178:Conserved
167:Intensive
156:Extensive
2614:Category
2531:Archived
2490:(2008).
2463:Archived
2356:Archived
2259:Archived
2203:22861139
2162:ProQuest
2095:Archived
2058:Archived
2013:24 March
2007:Archived
1806:Archived
1735:See also
1674:polygons
1662:Low poly
1364:function
1077:integral
1039:cylinder
1004:fuel oil
987:hogshead
947:teaspoon
896:imperial
801:pipettes
762:multiple
746:bladders
742:stomachs
419:pyramids
410:Elements
307:formulas
289:metonymy
261:imperial
183:yes for
2330:19 June
2042:Bibcode
1726:is the
1696:Density
1008:naphtha
777:biology
730:pinches
684:of the
682:measure
543:congius
506:In the
446:Liu Hui
423:spheres
377:frustum
369:cuboids
352:Pompeii
333:History
241:regions
237:measure
189:liquids
48:liquids
2620:Volume
2604:Volume
2567:
2502:
2429:
2399:
2389:
2224:
2201:
2191:
2164:
2141:
2131:
1982:
1974:
1964:
1910:
1900:
1676:. The
1472:, the
1349:region
1194:where
1051:height
1045:: the
1035:cuboid
1000:weight
983:bushel
969:barrel
965:gallon
933:drachm
868:of L.
789:spoons
744:, and
738:gourds
690:length
658:) and
652:second
624:pounds
615:gramme
522:, and
512:sester
281:length
269:gallon
233:Volume
197:plasma
195:, and
185:solids
134:barrel
106:gallon
58:, and
31:Volume
2466:(PDF)
2453:(PDF)
2423:(PDF)
2324:(PDF)
2311:(PDF)
2262:(PDF)
2249:(PDF)
2098:(PDF)
2085:(PDF)
1757:Notes
1043:prism
961:quart
929:minim
889:litre
854:metre
826:Units
680:As a
609:litre
603:stère
520:coomb
516:amber
458:India
401:digit
393:cubit
381:cones
315:Zero-
285:fluid
273:quart
257:litre
235:is a
193:gases
110:quart
98:Litre
2565:ISBN
2500:ISBN
2474:2022
2427:ISBN
2397:OCLC
2387:ISBN
2364:2022
2332:2016
2270:2022
2222:ISBN
2199:OCLC
2189:ISBN
2139:OCLC
2129:ISBN
2106:2022
2015:2009
1980:OCLC
1972:LCCN
1962:ISBN
1908:OCLC
1898:ISBN
1814:2022
1722:The
1711:The
1700:mass
1223:and
1089:line
1047:base
1037:and
1031:cube
979:peck
975:cord
957:gill
941:pint
834:and
803:and
787:and
779:and
752:and
694:area
692:and
674:and
650:and
582:and
524:seam
488:and
456:and
397:palm
379:and
363:and
327:four
321:and
319:one-
255:and
187:and
114:pint
52:cups
2050:doi
1715:or
1608:sin
1532:In
1476:is
1468:In
1354:in
898:or
813:bbl
641:of
350:in
295:).
263:or
243:in
239:of
161:yes
140:In
118:tsp
2616::
2529:.
2523:.
2455:.
2441:^
2409:^
2395:.
2348:.
2313:.
2286:.
2251:.
2197:.
2137:.
2087:.
2073:^
2056:.
2048:.
2038:42
2036:.
2032:.
2005:.
2001:.
1978:.
1970:.
1960:.
1920:^
1906:.
1896:.
1892:.
1872:^
1855:.
1830:.
1798:.
1785:^
1668:A
1053:.
1033:,
985:,
981:,
977:,
967:,
963:,
959:,
949:,
939:,
935:,
931:,
921:,
917:,
913:,
909:,
799:,
769:.
720:.
662:.
578:,
574:,
570:,
566:,
562:,
518:,
514:,
476:.
460:.
435:c.
399:,
395:,
375:,
371:,
317:,
275:,
271:,
172:no
132:,
130:yd
128:,
126:in
124:,
120:,
116:,
112:,
108:,
104:,
100:,
54:,
42:A
2573:.
2540:.
2508:.
2476:.
2435:.
2403:.
2366:.
2334:.
2296:.
2272:.
2242:"
2230:.
2205:.
2168:.
2145:.
2108:.
2067:.
2052::
2044::
2017:.
1986:.
1914:.
1866:.
1841:.
1816:.
1638:.
1632:d
1625:d
1618:d
1603:2
1593:D
1518:,
1515:z
1512:d
1505:d
1501:r
1498:d
1494:r
1489:D
1454:.
1451:z
1448:d
1444:y
1441:d
1437:x
1434:d
1430:1
1425:D
1400:1
1397:=
1394:)
1391:z
1388:,
1385:y
1382:,
1379:x
1376:(
1373:f
1352:D
1335:x
1332:d
1327:|
1323:)
1320:x
1317:(
1314:g
1308:)
1305:x
1302:(
1299:f
1295:|
1291:x
1286:b
1281:a
1270:2
1267:=
1264:V
1240:)
1237:x
1234:(
1231:g
1211:)
1208:x
1205:(
1202:f
1182:x
1179:d
1174:|
1168:2
1164:)
1160:x
1157:(
1154:g
1146:2
1142:)
1138:x
1135:(
1132:f
1128:|
1122:b
1117:a
1106:=
1103:V
989:.
971:;
953:;
943:;
925:;
433:(
225:L
180:?
169:?
158:?
149:m
74:V
62:.
23:.
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