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Girlfriends' Guide to Divorce
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Girlfriends' Guide to Divorce
Also known as
Girlfriends' Guide to Freedom (season 3)
Girlfriends' Guide to Bossing Up (season 4)
Girlfriends' Guide to Ever After (season 5)
Based on
Girlfriends' Guides series
by Vicki Iovine
Developed by
Marti Noxon
Starring
Lisa Edelstein
Beau Garrett
Necar Zadegan
Paul Adelstein
Janeane Garofalo
Alanna Ubach
Retta
Composer(s)
Robert Duncan
Country of origin
United States
Original language(s)
English
No. of seasons
5
No. of episodes
45 (list of episodes)
Production
Executive producer(s)
Marti Noxon
Vicki Iovine
Meryl Poster
Robert Duncan McNeill
Liz Kruger
Craig Shapiro
Camera setup
Single-camera
Running time
45 minutes
Production company(s)
Tiny Pyro Productions
Universal Cable Productions
Distributor
NBCUniversal Television Distribution
Release
Original network
Bravo
Picture format
480i (SDTV)
1080i (HDTV)
Original release
December 2, 2014 (2014-12-02) – July 19, 2018 (2018-07-19)
External links
Official website
Girlfriends' Guide to Divorce (also known as Girlfriends' Guide to Freedom in season 3, Girlfriends' Guide to Bossing Up in season 4, and Girlfriends' Guide to Ever After in season 5) is an American comedy-drama television series developed by Marti Noxon for the American cable network Bravo. Based on the Girlfriends' Guides book series by Vicki Iovine,[1] the series revolves around Abby McCarthy, a self-help author who finds solace in new friends and adventures as she faces an impending divorce. Lisa Edelstein portrays the main character Abby. Beau Garrett and Necar Zadegan co-star. Janeane Garofalo was part of the main cast for the first seven episodes of season 1 before departing the cast. She was replaced in episode 8 with Alanna Ubach. Paul Adelstein co-starred as a main cast member for the first two seasons, before being reduced to a recurring character for the third and fifth seasons. Retta recurred during the show's second season before being promoted to the main cast at the start of season 3.
Produced by Universal Cable Productions, it is the first original scripted series for Bravo.[2] A 13-episode first season was ordered by the network, which premiered on December 2, 2014.[3] The show debuted to 1.04 million viewers. Critical reception for the series has initially been generally positive, with particular praise towards Edelstein's performance and the series' quality over the reality series on Bravo. The show was eventually renewed for a second season, which premiered on December 1, 2015.[4] On April 13, 2016, it was announced that Bravo had renewed the show for a third, fourth and fifth season.[5] On August 5, 2016, it was announced that the fifth season would be the show's last.[6]
Contents
1 Cast
1.1 Main
1.2 Recurring
1.3 Guest
2 Episodes
3 Reception
3.1 Critical response
3.2 Ratings
4 Broadcast
4.1 Home media and streaming
5 References
6 External links
Cast[edit]
Main[edit]
Lisa Edelstein as Abby McCarthy, a self-help book author in her early 40s in Los Angeles. Her marriage has recently ended, throwing both her personal and professional lives into serious disarray.[1]
Beau Garrett as Phoebe Conte, a recent divorcée and former model. Up until now she has been a typical free spirit but she is now seeking a more stable life.[1]
Necar Zadegan as Delia Banai, a divorce attorney who represents Abby in her divorce. She struggles with career and personal life balance.[7]
Paul Adelstein as Jake Novak, Abby's husband from whom she is recently separated. He is a director with a limited amount of success and has up until now lived in Abby's shadow.[1] (seasons 1–2; recurring seasons 3 & 5)
Janeane Garofalo as Lyla Straley, Abby's friend, a divorcée and entertainment lawyer. A bitter custody battle with her ex causes her to flee California for Portland in the middle of season 1.[1] season 1, episodes 1–7)
Alanna Ubach as Jo Hernandez-Frumpkis, Abby's best friend from college who comes to live with her in LA. She is recently separated from her husband after finding out he had a second family.[8] (season 1, episode 8–season 5)
Retta as Barbara Sawyer, Abby's co-worker at SheShe, who later becomes her business partner. (seasons 3–5; recurring season 2)[9]
Recurring[edit]
Julianna Guill as Becca Riley, Jake's new girlfriend and a successful actress on a CW show
Patrick Heusinger as Max McCarthy, Abby's brother
J. August Richards as Ford, husband to Abby's brother
Warren Christie as Will, Abby's new boyfriend
Brian Markinson as Albert, Delia's boss and long-time fling. Delia has an affair with him while engaged to Gordon
Matthew Glave as Gordon Beech, Delia's client with whom she starts an affair
Brandon Jay McLaren as Marco, Phoebe's love interest
Aaron Staton as JD, Phoebe's love interest
Jean Smart as Katherine Miller, Abby's editor
Mark Valley as Dr. Harris, Abby's love interest
Maury Sterling as Rob Frumpkis, Jo's ex-husband
Will Kemp as Scott, the vegan confectioner at Jo's bakery
Megan Hilty as Charlene, Frumpkis' new girlfriend
Charles Mesure as Ralf, Phoebe's ex-husband
James Lesure as Mike Brady, Charlie's baseball coach
Daisy Betts as Gemma
Guest[edit]
Carrie Fisher as Cat
Suke Kaiser as Kori Wingo, Phoebe's former boss
Bernadette Peters as Annie
Laverne Cox as Adele Northrop
Victor Webster as Carl, a male escort who Abby spends the night talking to
Barry Bostwick as George, Abby's dad
Lesley Ann Warren as Dina, Abby's mom
Malcolm Jamal Warner as Darrell
Kathie Lee Gifford as herself
Hoda Kotb as herself
Kyle Richards as herself
Jessica Clark as Merete
Denise Richards as Temple Hampton, Will's new girlfriend
Episodes[edit]
Main article: List of Girlfriends' Guide to Divorce episodes
Season
Episodes
Originally aired
First aired
Last aired
1 13 December 2, 2014 (2014-12-02) February 24, 2015 (2015-02-24)
2 13 December 1, 2015 (2015-12-01) February 23, 2016 (2016-02-23)
3 7 January 11, 2017 (2017-01-11) February 22, 2017 (2017-02-22)
4 6 August 17, 2017 (2017-08-17) September 21, 2017 (2017-09-21)
5 6[10] June 14, 2018 (2018-06-14) July 19, 2018 (2018-07-19)[11]
Reception[edit]
Critical response[edit]
Edelstein and Garofalo's performances were praised by the critics.
Girlfriends' Guide to Divorce was met with generally positive reviews from television critics. At Metacritic, which assigns a weighted mean rating out of 100 to reviews from critics, the series received an average score of 69, based on 21 reviews. Lori Rackl of Chicago Sun-Times gave the episode a 4 star rating (out of 4 stars), calling it "a sometimes hilarious, sometimes heartbreaking story about an L.A.-based self-help author" and added that the first two episodes "reveal a much more nuanced, poignant tale, punctuated by some genuinely funny scenes."[12] LaToya Ferguson of The A.V. Club gave the series a grade of "A-", calling it "a very solid drama" that should be on HBO or Showtime. Ferguson also praised the characters and the series's messiness, writing "Visually, it's almost flawless (there's one obvious green-screen moment in the pilot, but it's not Ringer level), but every character here is deeply flawed."[13] Los Angeles Times's Mary McNamara lauded the series' cast's portrayal of the characters and deemed the series "smartly acted, crisply written and willing to address all manner of issues — marriage, betrayal, family economics, friendship, even the pitfalls of public domesticity — in gratifyingly complex ways."[14] Brian Lowry, writing for Variety, applauded the series' cast and material, noting how it sticks to the network's demographic while maintaining a level of quality.[15]
Gail Pennington of St. Louis Post-Dispatch called the series "a smart, solid examination of just how messy relationships are and how hard it is to make them work."[16] Slate's Willa Paskin highlighted Edelstein's portrayal of the lead character, describing her as "very well cast, both commanding and nurturing enough to seem like the ideal advice-giver" and noted that the series has "a satisfying and complex take on social dynamics in friendship and romance."[17] Alessandra Stanley of The New York Times praised Edelstein and Garofalo as "one reason" the show is entertaining and found the comic side of the series "a lot more fun."[18] Time's writer James Poniewozik praised the writing and Edelstein's "sympathetic" performance, noting that the latter "grounds a show that often otherwise plays like young-adult fiction for actual adults." However, Poniewozik opined that "there's one lesson Girlfriends’ Guide to Divorce has (over)learned from its Bravo peers: that there's no reality so compelling that it can’t be sweetened with a little Photoshop."[19] David Hinckley, writing for the New York Daily News, highlighted the series' best moments as those showing the messy side of marital discourse while heralding Edelstein's performance as "memorably moving."[20] Margaret Lyons of Vulture was critical of the several aspects of the series, including the characters Abby and Lyla's attitude on giving their spouses child support, but found the series to be its best "at its nastiest."[21]
Ratings[edit]
No. in
Episode title
Air date
Time slot (EST)
18–49 demo
rating
Viewership
(millions)
Ref
series
season
1 1 Rule No. 23: Never Lie to the Kids December 2, 2014 Tuesdays
10:00 P.M. 0.5 1.04 [22]
2 2 Rule No. 174: Never Trust Anyone Who Charges By the Hour December 9, 2014 0.4 0.85 [23]
3 3 Rule No. 47: Always Take Advantage of "Me" Time December 16, 2014 0.3 0.70 [24]
4 4 Rule No. 426: Fantasyland: A Great Place to Visit December 23, 2014 0.3 0.77 [25]
5 5 Rule No. 21: Leave Childishness to Children December 30, 2014 0.4 0.87 [26]
6 6 Rule No 33: When in Doubt, Run Away January 6, 2015 0.4 0.87 [27]
7 7 Rule No. 67: Don't Kill the Princess January 13, 2015 0.4 0.77 [28]
8 8 Rule No. 17: Ask the Answer Lady January 20, 2015 0.3 0.69 [29]
9 9 Rule No. 32: F-You, Rob Frumpkis January 27, 2015 0.4 0.91 [30]
10 10 Rule No. 3: Don't Stand in the Doorway February 3, 2015 0.3 0.74 [31]
11 11 Rule No. 46: Keep the Holidays Low Key February 10, 2015 0.4 0.83 [32]
12 12 Rule No. 92: Don't Do the Crime If You Can't Do the Time February 17, 2015 0.4 0.74 [33]
13 13 Rule No. 101: Know When It's Time to Move On February 24, 2015 0.3 0.79 [34]
Broadcast[edit]
Girlfriends' Guide to Divorce premiered in the United States on Bravo on December 2, 2014, in Canada on Slice on January 9, 2015,[35] and in the United Kingdom on Lifetime UK on September 15, 2015.[36]
Home media and streaming[edit]
The first season was released on DVD in region 1 on October 13, 2015.[37] On November 1, 2015 [38] season 1 of Girlfriends' Guide to Divorce became available to stream in the US for Netflix subscribers.[39] The show is also available from electronic sell-through platforms such as iTunes,[40] Amazon Instant Video,[41] and Vudu.[42]
References[edit]
^ Jump up to: a b c d e Bibel, Sara (February 19, 2014). "Bravo Orders 13 Episodes of Scripted Drama 'Girlfriends' Guide to Divorce' Starring Lisa Edelstein". Zap2it. Retrieved February 20, 2014.
Jump up ^ Andreeva, Nellie (February 19, 2014). "Bravo Picks Up Marti Noxon's 'Girlfriends' Guide To Divorce' As First Scripted Series". Deadline Hollywood. Retrieved February 20, 2014.
Jump up ^ Kondolojy, Amanda (July 14, 2014). "'Million Dollar Listing New York' Renewed for Season Four + Bravo Greenlights Three New Series". TV by the Numbers (Press release). Archived from the original on July 4, 2015. Retrieved July 14, 2014.
Jump up ^ Dixon, Dani (September 30, 2015). "Season 2 of Bravo's Scripted Series 'Girlfriend's Guide To Divorce' To Premiere December 1". TV by the Numbers (Press release). Tribune Digital Ventures. Retrieved September 30, 2015.
Jump up ^ "'Girlfriends' Guide to Divorce' gets 3-season renewal at Bravo". TV By The Numbers by zap2it.com. Retrieved 2016-04-13.
Jump up ^ "Bravo's 'Girlfriends' Guide to Divorce' to End After Season 5 (Exclusive)". The Hollywood Reporter. 2016-08-05. Retrieved 2018-01-03.
Jump up ^ Andreeva, Nellie (March 14, 2014). "Bravo's 'Girlfriend's Guide To Divorce' Adds 'Rake's Necar Zadegan As New Regular". Deadline Hollywood. Retrieved August 14, 2014.
Jump up ^ "Alanna Ubach Joins Bravo's 'Girlfriends Guide To Divorce', Matthew Glave & Virginia Williams To Recur". Deadline Hollywood. August 13, 2014. Retrieved August 14, 2014.
Jump up ^ Ausiello, Michael. "Girlfriends' Guide to Divorce Promotes Retta to Series Regular in Season 3". TV Line. TVLine Media, LLC. Retrieved June 30, 2016.
Jump up ^ Sandberg, Bryn Elise (August 5, 2016). "Bravo's 'Girlfriends' Guide to Divorce' to End After Season 5 (Exclusive)". The Hollywood Reporter.
Jump up ^ "GIRLFRIENDS' GUIDE TO DIVORCE (BRAVO)". The Futun Critic. Retrieved June 21, 2018.
Jump up ^ Rackl, Lori (November 30, 2014). "Bravo found a keeper with great new series 'Girlfriends' Guide to Divorce'". Chicago Sun-Times. Chicago: Tim Knight. Sun-Times Media Group. Archived from the original on 31 January 2015. Retrieved 6 February 2015.
Jump up ^ Ferguson, LaToya (December 2, 2014). "Girlfriends' Guide To Divorce is Bravo's beautifully messy introduction to scripted drama". The A.V. Club. Chicago: The Onion, Inc. Retrieved 6 February 2015.
Jump up ^ McNamara, Mary (December 2, 2014). "Review 'Girlfriends' Guide to Divorce' marries Bravo to good TV". Los Angeles Times. Los Angeles: Austin Beutner. Tribune Publishing. Retrieved 6 February 2015.
Jump up ^ Lowry, Brian (November 30, 2014). "TV Review: Bravo's 'Girlfriends' Guide To Divorce'". Variety. Los Angeles. Penske Business Media. Retrieved 8 February 2015.
Jump up ^ Pennington, Gail (December 2, 2014). "TV review: Bravo's 'Girlfriends' Guide' is surprisingly good". St. Louis Post-Dispatch. St. Louis: Ray Farris. Lee Enterprises. Retrieved 8 February 2015.
Jump up ^ Paskin, Willa (December 2, 2014). "Lowbrow and Delectable". Slate. The Slate Group. Retrieved 9 February 2015.
Jump up ^ Stanley, Alessandra (December 1, 2014). "Divvying Up Perks, Pity, Scorn and Snark". The New York Times. New York City: Dean Baquet. The New York Times Company. Retrieved 9 February 2015.
Jump up ^ Poniewozik, James (December 1, 2014). "Review: The Image-Conscious Uncoupling of Girlfriends' Guide to Divorce". Time. New York City: Time Inc. Retrieved 24 February 2015.
Jump up ^ Roush, Matt (December 2, 2014). "Tuesday TV: One for the Family (Toy Story) and One for Adults (Girlfriends' Guide)". TV Guide. Radnor: CBS Interactive. Retrieved 24 February 2015.
Jump up ^ Lyons, Margaret (December 2, 2014). "Girlfriends' Guide to Divorce, Bravo's First Scripted Series, Is At Its Best When It's At Its Nastiest". New York. New York City: New York Media, LLC. Retrieved 25 February 2015.
Jump up ^ Metcalf, Mitch (December 3, 2014). "SHOWBUZZDAILY's Top 25 Tuesday Cable Originals (& Network Update): 12.2.2014". Showbuzz Daily. Retrieved 23 January 2015.
Jump up ^ Bibel, Sara (December 10, 2014). "Tuesday Cable Ratings: 'Sons of Anarchy' Wins Night, 'Anarchy Afterword', 'Pretty Little Liars', 'Benched', 'Ground Floor' & More". TV by the Numbers. Tribune Media Services. Retrieved 1 September 2014.
Jump up ^ Kondolojy, Amanda (December 17, 2014). "Tuesday Cable Ratings: 'Ink Master' Tops Night + 'The Real Housewives of Beverly Hills', 'The Curse of Oak Island', NBA Basketball & More". TV by the Numbers. Tribune Media Services. Retrieved 1 September 2014.
Jump up ^ Metcalf, Mitch (December 24, 2014). "SHOWBUZZDAILY's Top 25 Tuesday Cable Originals (& Network Update): 12.23.2014". Showbuzz Daily. Retrieved 24 December 2014.
Jump up ^ Kondolojy, Amanda (January 2, 2015). "Tuesday Cable Ratings: College Football Tops Night + 'Moonshiners' The Little Couple', 'Ground Floor' & More". TV by the Numbers. Tribune Media Services. Retrieved 3 January 2015.
Jump up ^ Bibel, Sara (January 8, 2015). "Tuesday Cable Ratings: 'The Haves and the Have Nots' Wins Night' 'Pretty Little Liars', 'Switched at Birth', 'Dance Moms', 'The Real World' & More". TV by the Numbers. Tribune Media Services. Retrieved 10 January 2015.
Jump up ^ The Futon Critic Staff (July 13, 2011). "Monday's Cable Ratings: "Pawn Stars" Tops Heated Race". The Futon Critic. Retrieved 1 September 2014.
Jump up ^ The Futon Critic Staff (July 20, 2011). "Monday's Cable Ratings: History's "Pawn Stars" Holds Onto Crown". The Futon Critic. Retrieved 1 September 2014.
Jump up ^ Kondolojy, Amanda (January 28, 2015). "Tuesday Cable Ratings: 'The Real Housewives of Beverly Hills' Tops Night + 'Pretty Little Liars', 'The Haves and the Have Nots', 'Sons of Liberty' & More". TV by the Numbers. Tribune Media Services. Retrieved 6 February 2015.
Jump up ^ Bibel, Sara (February 4, 2015). "Tuesday Cable Ratings: 'Being Mary Jane' Wins Night, 'Moonshiners', 'The Haves and the Have Nots', 'Pretty Little Liars', 'Justified' & More". TV by the Numbers. Tribune Media Services. Retrieved 6 February 2015.
Jump up ^ Kondolojy, Amanda (February 11, 2015). "Tuesday Cable Ratings: 'The Haves and the Have Nots' Tops Night + 'Being Mary Jane', 'Pretty Little Liars', 'Real Housewives of Beverly Hills' & More". TV by the Numbers. Tribune Media Services. Retrieved 24 February 2015.
Jump up ^ Bibel, Sara (February 19, 2015). "Tuesday Cable Ratings: '19 Kids and Counting' Wins Night, 'Being Mary Jane', 'Pretty Little Liars', 'Rizzoli & Isles', 'Cougar Town' & More". TV by the Numbers. Tribune Media Services. Retrieved 24 February 2015.
Jump up ^ Metcalf, Mitch (February 25, 2015). "SHOWBUZZDAILY's Top 25 Tuesday Cable Originals (& Network Update): 2.24.2015". Showbuzz Daily. Retrieved 27 February 2015.
Jump up ^ "Girlfriends' Guide to Divorce to premiere on Slice in Canada". The TV Watercooler. http://google.com/+JeevanBrar. Retrieved 2015-11-10. External link in |publisher= (help)
Jump up ^ "Lifetime UK Acquires 'Girlfriends' Guide To Divorce', Sets September Premiere - TVWise". TVWise. Retrieved 2015-11-10.
Jump up ^ Girlfriends' Guide to Divorce: Season 1, 2015-10-13, retrieved 2015-11-10
Jump up ^ Armstrong, Olivia (2015-10-30). "What's New On Netflix This Weekend: 'Manson Family Vacation', November Titles, And More". Decider | Where To Stream Movies & Shows on Netflix, Hulu, Amazon Instant, HBO Go. Retrieved 2015-11-10.
Jump up ^ "Girlfriends' Guide to Divorce | Netflix". www.netflix.com. Retrieved 2015-11-10.
Jump up ^ "Girlfriends' Guide to Divorce, Season 1 on iTunes". iTunes. Retrieved 2015-11-10.
Jump up ^ "Watch Girlfriends' Guide to Divorce Season 1 Episode - Amazon Video". www.amazon.com. Retrieved 2015-11-10.
Jump up ^ "VUDU - Watch Movies". www.vudu.com. Retrieved 2015-11-10.
External links[edit]
Official website
Girlfriends' Guide to Divorce on IMDb
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Volume
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
For other uses, see Volume (disambiguation).
Volume
A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
Common symbols
V
SI unit
Cubic metre [m3]
Other units
Litre, Fluid ounce, gallon, quart, pint, tsp, fluid dram, in3, yd3, barrel
In SI base units
1 m3
Dimension
L3
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.[1] Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.[2]
In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.
Contents
1 Units
2 Related terms
3 Volume in calculus
4 Volume formulas
4.1 Volume ratios for a cone, sphere and cylinder of the same radius and height
4.2 Volume formula derivations
4.2.1 Sphere
4.2.2 Cone
4.2.3 Polyhedron
5 Volume in differential geometry
6 Volume in thermodynamics
7 See also
8 References
9 External links
Units[edit]
Volume measurements from the 1914 The New Student's Reference Work.
Approximate conversion to metric (mL)[3]
Imp.
U.S.
Liquid
Dry
Gill 142 118 138
Pint 568 473 551
Quart 1137 946 1101
Gallon 4546 3785 4405
Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm3) is the volume of a cube whose sides are one centimetre (1 cm) in length.
In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,
so
1 cubic metre = 1000 litres.
Small amounts of liquid are often measured in millilitres, where
1 millilitre = 0.001 litres = 1 cubic centimetre.
In the same way, large amounts can be measured in megalitres, where
1 million litres = 1000 cubic metres = 1 megalitre.
Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.
See also: unusual and obsolete units of volume
Related terms[edit]
Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, and to liquids, grain, or the like, which take the shape of that which holds them".[4] (The word capacity has other unrelated meanings, as in e.g. capacity management.) Capacity is not identical in meaning to volume, though closely related; the capacity of a container is always the volume in its interior. Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are closely related: one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the capacity of a vehicle's fuel tank is rarely stated in cubic feet, for example, but in gallons (an imperial gallon fills a volume of 0.1605 cu ft).
The density of an object is defined as the ratio of the mass to the volume.[5] The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied.
The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s−1]).
Volume in calculus[edit]
Further information: Volume element
In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} and is usually written as:
∭ D 1 d x d y d z . {\displaystyle \iiint \limits _{D}1\,dx\,dy\,dz.}
The volume integral in cylindrical coordinates is
∭ D r d r d θ d z , {\displaystyle \iiint \limits _{D}r\,dr\,d\theta \,dz,}
and the volume integral in spherical coordinates (using the convention for angles with θ {\displaystyle \theta } as the azimuth and ϕ {\displaystyle \phi } measured from the polar axis; see more on conventions) has the form
∭ D ρ 2 sin  ϕ d ρ d θ d ϕ . {\displaystyle \iiint \limits _{D}\rho ^{2}\sin \phi \,d\rho \,d\theta \,d\phi .}
Volume formulas[edit]
Shape
Volume formula
Variables
Cube a 3 {\displaystyle a^{3}\;} a = length of any side (or edge)
Circular Cylinder π r 2 h {\displaystyle \pi r^{2}h\;} r = radius of circular base, h = height
Prism B h {\displaystyle Bh} B = area of the base, h = height
Cuboid l w h {\displaystyle lwh} l = length, w = width, h = height
Triangular prism 1 2 b h l {\displaystyle {\frac {1}{2}}bhl} b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases
Triangular prism (with given lengths of three sides) 1 4 h − a 4 + 2 ( a b ) 2 + 2 ( a c ) 2 − b 4 + 2 ( b c ) 2 − c 4 {\displaystyle {\frac {1}{4}}h{\sqrt {-a^{4}+2(ab)^{2}+2(ac)^{2}-b^{4}+2(bc)^{2}-c^{4}}}} a, b, and c = lengths of sides
h = height of the triangular prism
Sphere 4 3 π r 3 = 1 6 π d 3 {\displaystyle {\frac {4}{3}}\pi r^{3}={\frac {1}{6}}\pi d^{3}} r = radius of sphere
d = diameter of sphere
which is the integral of the surface area of a sphere
Ellipsoid 4 3 π a b c {\displaystyle {\frac {4}{3}}\pi abc} a, b, c = semi-axes of ellipsoid
Torus ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 {\displaystyle \left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}} r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus)
Pyramid 1 3 B h {\displaystyle {\frac {1}{3}}Bh} B = area of the base, h = height of pyramid
Square pyramid 1 3 s 2 h {\displaystyle {\frac {1}{3}}s^{2}h\;} s = side length of base, h = height
Rectangular pyramid 1 3 l w h {\displaystyle {\frac {1}{3}}lwh} l = length, w = width, h = height
Cone 1 3 π r 2 h {\displaystyle {\frac {1}{3}}\pi r^{2}h} r = radius of circle at base, h = distance from base to tip or height
Regular tetrahedron[6] 2 12 a 3 {\displaystyle {{\sqrt {2}} \over 12}a^{3}\,} Edge length, a
Parallelepiped a b c K {\displaystyle abc{\sqrt {K}}}
K = 1 + 2 cos  ( α ) cos  ( β ) cos  ( γ ) − cos 2  ( α ) − cos 2  ( β ) − cos 2  ( γ ) {\displaystyle {\begin{aligned}K=1&+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\end{aligned}}}
a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges
Any volumetric sweep
(calculus required) ∫ a b A ( h ) d h {\displaystyle \int _{a}^{b}A(h)\,\mathrm {d} h} h = any dimension of the figure, A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep.
(This will work for any figure if its cross-sectional area can be determined from h).
Any rotated figure (washer method;
calculus required) π ∫ a b ( [ R O ( x ) ] 2 − [ R I ( x ) ] 2 ) d x {\displaystyle \pi \int _{a}^{b}\left({\left[R_{O}(x)\right]}^{2}-{\left[R_{I}(x)\right]}^{2}\right)\mathrm {d} x} R O {\displaystyle R_{O}} and R I {\displaystyle R_{I}} are functions expressing the outer and inner radii of the function, respectively.
Volume ratios for a cone, sphere and cylinder of the same radius and height[edit]
A cone, sphere and cylinder of radius r and height h
The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.
Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is
1 3 π r 2 h = 1 3 π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 1 , {\displaystyle {\frac {1}{3}}\pi r^{2}h={\frac {1}{3}}\pi r^{2}\left(2r\right)=\left({\frac {2}{3}}\pi r^{3}\right)\times 1,}
the volume of the sphere is
4 3 π r 3 = ( 2 3 π r 3 ) × 2 , {\displaystyle {\frac {4}{3}}\pi r^{3}=\left({\frac {2}{3}}\pi r^{3}\right)\times 2,}
while the volume of the cylinder is
π r 2 h = π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 3. {\displaystyle \pi r^{2}h=\pi r^{2}(2r)=\left({\frac {2}{3}}\pi r^{3}\right)\times 3.}
The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.[7]
Volume formula derivations[edit]
Sphere[edit]
The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.
The surface area of the circular disk is π r 2 {\displaystyle \pi r^{2}} .
The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is
y = r 2 − x 2 {\displaystyle y={\sqrt {r^{2}-x^{2}}}}
or
z = r 2 − x 2 {\displaystyle z={\sqrt {r^{2}-x^{2}}}}
where y or z can be taken to represent the radius of a disk at a particular x value.
Using y as the disk radius, the volume of the sphere can be calculated as
∫ − r r π y 2 d x = ∫ − r r π ( r 2 − x 2 ) d x . {\displaystyle \int _{-r}^{r}\pi y^{2}\,dx=\int _{-r}^{r}\pi \left(r^{2}-x^{2}\right)\,dx.}
Now
∫ − r r π r 2 d x − ∫ − r r π x 2 d x = π ( r 3 + r 3 ) − π 3 ( r 3 + r 3 ) = 2 π r 3 − 2 π r 3 3 . {\displaystyle \int _{-r}^{r}\pi r^{2}\,dx-\int _{-r}^{r}\pi x^{2}\,dx=\pi \left(r^{3}+r^{3}\right)-{\frac {\pi }{3}}\left(r^{3}+r^{3}\right)=2\pi r^{3}-{\frac {2\pi r^{3}}{3}}.}
Combining yields V = 4 3 π r 3 . {\displaystyle V={\frac {4}{3}}\pi r^{3}.}
This formula can be derived more quickly using the formula for the sphere's surface area, which is 4 π r 2 {\displaystyle 4\pi r^{2}} . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to
∫ 0 r 4 π r 2 d r = 4 3 π r 3 . {\displaystyle \int _{0}^{r}4\pi r^{2}\,dr={\frac {4}{3}}\pi r^{3}.}
Cone[edit]
The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.
However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0, 0, 0) with radius r, is as follows.
The radius of each circular disk is r if x = 0 and 0 if x = h, and varying linearly in between—that is,
r h − x h . {\displaystyle r{\frac {h-x}{h}}.}
The surface area of the circular disk is then
π ( r h − x h ) 2 = π r 2 ( h − x ) 2 h 2 . {\displaystyle \pi \left(r{\frac {h-x}{h}}\right)^{2}=\pi r^{2}{\frac {(h-x)^{2}}{h^{2}}}.}
The volume of the cone can then be calculated as
∫ 0 h π r 2 ( h − x ) 2 h 2 d x , {\displaystyle \int _{0}^{h}\pi r^{2}{\frac {(h-x)^{2}}{h^{2}}}dx,}
and after extraction of the constants
π r 2 h 2 ∫ 0 h ( h − x ) 2 d x {\displaystyle {\frac {\pi r^{2}}{h^{2}}}\int _{0}^{h}(h-x)^{2}dx}
Integrating gives us
π r 2 h 2 ( h 3 3 ) = 1 3 π r 2 h . {\displaystyle {\frac {\pi r^{2}}{h^{2}}}\left({\frac {h^{3}}{3}}\right)={\frac {1}{3}}\pi r^{2}h.}
Polyhedron[edit]
Main article: Volume of a polyhedron
Volume in differential geometry[edit]
Main article: Volume form
In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form of top degree (i.e., whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. Integrating the volume form gives the volume of the manifold according to that form.
An oriented pseudo-Riemannian manifold has a natural volume form. In local coordinates, it can be expressed as
ω = | g | d x 1 ∧ ⋯ ∧ d x n , {\displaystyle \omega ={\sqrt {|g|}}\,dx^{1}\wedge \dots \wedge dx^{n},}
where the d x i {\displaystyle dx^{i}} are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold, and g {\displaystyle g} is the determinant of the matrix representation of the metric tensor on the manifold in terms of the same basis.
Volume in thermodynamics[edit]
Main article: Volume (thermodynamics)
In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit of mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law.
See also[edit]
Area
Banach–Tarski paradox
Conversion of units
Dimensional weight
Dimensioning
Length
Measure
Orders of magnitude (volume)
Perimeter
Volume (thermodynamics)
Volumography
Weight
References[edit]
Jump up ^ "Your Dictionary entry for "volume"". Retrieved 2010-05-01.
Jump up ^ One litre of sugar (about 970 grams) can dissolve in 0.6 litres of hot water, producing a total volume of less than one litre. "Solubility". Retrieved 2010-05-01. Up to 1800 grams of sucrose can dissolve in a liter of water.
Jump up ^ "General Tables of Units of Measurement". NIST Weights and Measures Division. Archived from the original on 2011-12-10. Retrieved 2011-01-12.
Jump up ^ "capacity". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library membership required.)
Jump up ^ "density". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library membership required.)
Jump up ^ Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i).
Jump up ^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Retrieved 2007-01-02.
External links[edit]
Wikimedia Commons has media related to Volume.
Perimeters, Areas, Volumes at Wikibooks
Volume at Wikibooks
Retrieved from "https://en.wikipedia.org/w/index.php?title=Volume&oldid=865559877"
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Volume of a polyhedron

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Orientation (mathematics)

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Extensive parameter

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Intensive property

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Function of state

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http://www.yourdictionary.com/volume

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H. S. M. Coxeter

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Volum – Catalan
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Volumen – Spanish
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Volum – Norwegian
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Volum – Norwegian Nynorsk
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Volum – Occitan
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Volume – Portuguese
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P'ulin – Quechua
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Vollum – Scots
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Volym – Swedish
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Hacim – Turkish
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