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The Night Shift (TV series)
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This article is about the NBC series. For the General Hospital spinoff, see General Hospital: Night Shift. For other television series, see Night Shift (disambiguation).
The Night Shift
Genre
Medical drama
Created by
Gabe Sachs
Jeff Judah
Starring
Eoin Macken
Jill Flint
Ken Leung
Brendan Fehr
Robert Bailey, Jr.
Jeananne Goossen
JR Lemon
Freddy Rodriguez
Daniella Alonso
Scott Wolf
Tanaya Beatty
Composer(s)
Fred Coury
Country of origin
United States
Original language(s)
English
No. of seasons
4
No. of episodes
45 (list of episodes)
Production
Executive producer(s)
Gabe Sachs
Jeff Judah
Producer(s)
Zach Lutsky
Jill Danton
Location(s)
Albuquerque, New Mexico
San Antonio, Texas (setting)
Cinematography
Lex duPont
Arthur Albert
Running time
43 minutes
Production company(s)
Sachs/Judah Productions
Sony Pictures Television
Release
Original network
NBC
Picture format
1080i (16:9 HDTV)
Original release
May 27, 2014 (2014-05-27) – August 31, 2017 (2017-08-31)
External links
Official website
www.nbc.com/the-night-shift
The Night Shift is an American medical drama series that premiered on May 27, 2014, on NBC.[1][2] The series was created by Gabe Sachs and Jeff Judah, and follows the lives of the staff who work the late night shift in the emergency room at San Antonio Memorial Hospital.
On November 17, 2016, NBC renewed the series for a fourth season, which premiered on June 22, 2017.[3][4]
Contents
[hide]
1 Synopsis
2 Cast and characters
2.1 Main
2.1.1 Former main characters
2.2 Recurring
3 Episodes
4 Production
4.1 Development
4.2 Casting
4.3 Filming
5 Reception
5.1 Critical reception
5.2 Ratings
6 References
7 External links
Synopsis[edit]
The series follows the overnight shift at San Antonio Memorial Hospital, where three of the doctors have a connection to the U.S. military. Dr. TC Callahan is an former-Army medic who initially exhibits PTSD-type symptoms, having watched his brother die right in front of him on the battlefield. He frequently breaks rules and butts heads with his ex-girlfriend and newly-appointed head of the night shift, Dr. Jordan Alexander, and the hospital's administrator, Michael Ragosa. Dr. Topher Zia is a private-sector surgeon who previously spent a lot of time on the battlefield as an Army Ranger, while Dr. Drew Alister is a gay Army medic still active in the reserves who initially tries to hide his sexual orientation for fear of backlash. Dr. Krista Bell-Hart is a young surgical resident trying to work her way up the ranks, while fellow resident Dr. Paul Cummings works hard to emerge from the shadow of his father, a famous surgeon at Johns Hopkins. The shift's lone psychiatrist, Dr. Landry de la Cruz, briefly works on the night shift.
Jordan is initially in a relationship with Dr. Scott Clemmens, which complicates things when Scott becomes the Chief of Surgery at the hospital and sees that Jordan still has feelings for TC. After a cancer scare, Ragosa finds that he can no longer tolerate the stresses of his administrator job, and he quits that position in order to pursue his original dream of becoming a doctor. Ragosa passes exams, becoming a doctor just before his ex-wife moves his family to Dallas, and he follows. In season 3, Jordan hires new med-school graduate Dr. Shannon Rivera as a replacement.
In the season 3 finale, Paul's father returns to the hospital to buy it before a plastic surgery company tears it down. Despite this, he fires Topher due to his way of running the hospital, and most of the staff quits in protest. In the season 4 premiere, the staff returns soon after, save for Topher after he and his daughter are killed in a car accident.
Cast and characters[edit]
Main[edit]
Eoin Macken as Dr. Thomas Charles "TC" Callahan: A former army medic and Ranger from Baltimore who witnessed his brother Thad's death in Afghanistan, and immediately had to decide to donate his heart to a fellow soldier[5]. After being discharged, he returned with PTSD and his relationship with Jordan ended. In the first season, his recklessness and sports betting put him in trouble, especially with Ragosa, and was secretly dating Landry. In the second season, he rekindles his relationship with Jordan[6], but they break up again after she miscarries[7]. In the third season, he dates Jessica Sanders until he learns of her involvement in the sale of the hospital. Afterwards, he leaves on a humanitarian mission with Syd to Syria, which gets complicated quickly.
Jill Flint as Dr. Jordan Alexander: TC's ex-girlfriend. In the first season, she becomes Head of the ER night shift[8] and dates Scott Clemmens, but she still has feelings for TC. In season two, she renounces the position and rekindles her relationship with TC, but they break up after she miscarries in the second season finale.[7] In season four, she becomes Head of the ER night shift again.
Brendan Fehr as Dr. Andrew "Drew" Alister: An army medic still active in the reserves and chief resident. He was a nurse until the army paid for his medical education[8]. He tried to hide his homosexuality until he outed himself when boyfriend Rick, also in the army, had an accident, and Scott was forced to amputate half of his leg. Krista convinced him to be brave for Rick.[9] After a fight and temporary breakup, Drew and Rick got married in the second season. He returns to active service early in the third season, but his actions to save a young 15 year-old child bride results in an unwanted early end of deployment. He and Rick later adopt an orphan girl, Brianna.
Robert Bailey Jr. as Dr. Paul Cummings: a first year surgical resident and Columbia[8] graduate who works hard to emerge from the shadow of his famous father. He is often the butt of the practical jokes by the ER staff.
JR Lemon as Nurse Kenny Fournette: An ex-college football star who became a nurse after an injury ended his sports career. He has dated multiple staff members and colleagues, including Krista, Gwen, Nurse Diaz and Dr. Bella Cummings (Paul's sister).
Scott Wolf as Dr. Scott Clemmens: (recurring seasons 1–2, main season 3–) A trauma surgeon and a recovering alcoholic, he becomes Head of Trauma Surgery in the first season.[10] He dated Jordan in the first season then TC's sister-in-law Annie in the third season.
Tanaya Beatty as Dr. Shannon Rivera: (season 3–) A Native-American doctor whom Jordan recruits to work the Night Shift. She grew up in Foster care. Her initial adversarial relationship with Paul flourishes into a romance.
Former main characters[edit]
Ken Leung as Dr. Christopher "Topher" Zia: (season 1–3)[11] A former army medic and Ranger and TC's longtime friend, to the point his oldest daughter calls him "Uncle TC"[10]. Married with a daughter, his wife gave birth to twin daughters during a storm[10]. In season 2, he becomes head of the ER night shift. In the season 3 finale[12], he is fired by the new E.R. owner, Dr. Julian Cummings, for being too willing to give high-priced treatments to people who can't pay. In the second episode of the fourth season it is revealed that Topher, along with his oldest daughter, were killed in a car accident (offscreen) after being hit by a drunk driver.
Jeananne Goossen as Dr. Krista Bell-Hart: (seasons 1–2)[13] A first year surgical resident. She had a crush on Drew until she learned of his homosexuality,[14] but became good friends afterwards. She later dated Kenny and Joey Chavez.
Freddy Rodriguez as Dr. Michael Ragosa: (seasons 1–2)[15] In the first season, he was the newly appointed hospital administrator[8]. Recently separated from his wife, who took their children with her[5] and whom he later divorced[16], he also suffered from a relapsed tumor that was affecting his vision. In the second season, after his brain cancer surgery, he quit his position to finish his medical studies[6] and became a resident at the hospital. In the third season, it's revealed that he is now a resident in Dallas after his ex-wife moved their children there.
Daniella Alonso as Dr. Landry de la Cruz: (season 1) A hospital psychiatrist who dated TC until she realized he still had feelings for Jordan.[10]
Recurring[edit]
Esodie Geiger as Nurse Molly Ramos (seasons 1–4).
Alma Sisneros as Nurse Jocelyn Diaz (seasons 1–4).
Catharine Pilafas as Nurse Bardocz (seasons 1–3).
Luke Macfarlane as Rick Lincoln[17]. Drew's army captain boyfriend. While coming home from deployment in Afghanistan, his bus had an accident and Scott was forced to amputate his right leg under the knee[9]. He later becomes a combat trainer. After a fight and temporary breakup, he becomes Drew's husband in the season 2 finale[7] and they adopt an orphaned girl, Brianna, in season 3.
Merle Dandridge as EMT Gwen Gaskin (seasons 2–3).[18] Jordan's friend who dates Kenny until early in season 3, when she leaves to study law at Georgetown.
Sarah Jane Morris as Annie Callahan (seasons 2–3; guest season 4). TC's sister-in-law, the widow of his brother Thad. She is a recovering drug addict. In the third season, she dates Scott. In the third episode of season four, she commits suicide by jumping off of a bridge.
Adam Rodriguez as Dr. Joey Chavez (season 2). An eccentric surgeon with a tragic past. A former USAF doctor[6] who lost his son.
James McDaniel as Dr. Julian Cummings (season 2–). Paul's father and a renowned neurosurgeon. His company buys the E.R. in the third season finale.
Briana Marin as Nurse Nina Alvarez (season 3–).
Jennifer Beals[19] as Dr. Sydney "Syd" Jennings (season 3; guest season 4). A trauma doctor and Army commander. Drew's superior in Afghanistan, she gets the brunt of his actions and is forced to remain when she was hoping to return to see her daughter. After returning stateside, she decides to move to D.C. to avoid a custody battle with her ex-husband, who's moving there with his new wife. She leaves in a humanitarian mission in Syria with TC not long afterwards. After being captured and nearly killed at the start of season four, she tells TC that she is returning to the U.S. to be with her daughter.
AnnaLynne McCord[20] as Jessica Sanders (season 3), a pharmaceutical rep who is secretly involved in the sale of the E.R. to an insurance company. She briefly dates TC.
Kyla Kenedy as Brianna (season 3–). A young teen orphaned in an accident that claims her mother's life. After receiving a needed lung transplant, she is adopted by Drew and Rick.
Elizabeth Sung as Sumei Zia (season 3), Topher's mother.
Rana Roy as Dr. Amira (season 4–)[21]. A civilian volunteer who rescues TC after the bomb explosion that hits Syd and his camp site. After she and TC sleep together, she tells him she is still married, though in the process of divorcing her husband who is in England. She shows up in Texas briefly before returning to Syria.
Mark Consuelos as Dr. Cain Diaz (season 4)[22]. A confident and aggressive nurse who emigrated to the United States from Mexico. It is later revealed that he was a doctor in Mexico, and after being fired for overstepping his duties as a nurse, Jordan hires him as a resident doctor. It is later revealed that he was on the run and he quits working for the hospital with no notice.
Erica Tazel as Dr. Bella Cummings (season 4). Paul's urologist sister.
Episodes[edit]
Main article: List of The Night Shift episodes
Season
Episodes
Originally aired
First aired
Last aired
1 8 May 27, 2014 (2014-05-27) July 15, 2014 (2014-07-15)
2 14 February 23, 2015 (2015-02-23) May 18, 2015 (2015-05-18)
3 13 June 1, 2016 (2016-06-01) August 31, 2016 (2016-08-31)
4 10 June 22, 2017 (2017-06-22) August 31, 2017 (2017-08-31)
Production[edit]
Development[edit]
The series first appeared as part of NBC development slate in October 2011. However, NBC decided to not go forward with a pilot order.[23] In August 2012, NBC decided to revisit the pilot script for the series, then known as The Last Stand.[24] On October 8, 2012, NBC placed a pilot order, with the new name After Hours.[25] The pilot was directed by Pierre Morel and written by Gabe Sachs and Jeff Judah.
On April 18, 2013, NBC ordered four additional scripts under a third and final title, The Night Shift.[26] On May 10, 2013, NBC officially ordered The Night Shift to series.[27]
Casting[edit]
Casting announcements began in October 2012, with Eoin Macken first cast in the role of TC Callahan, a doctor who has recently returned from the army who constantly disagrees with his superiors and does things his own way.[28] Freddy Rodriguez was the next actor cast in the series, in the role of Michael Ragosa, the hospital's administrator who originally wanted to be a doctor.[29] Ken Leung and Jeananne Goossen were then added to the cast, with Leung cast in the role of Topher, an emergency room doctor who previously helped soldiers that were injured in battle. Goossen signed onto the role of Krista, a beautiful resident at the hospital.[30] In early November, Robert Bailey Jr. joined the series as Paul Cummings, a young, but squeamish resident at the hospital.[31] Jill Flint later signed onto the role of Jordan Alexander, the newly promoted Chief of the Night Shift, who once dated T.C.[32] Daniella Alonso was the last actor cast in the series. Alonso played the role of Dr. Landry de la Cruz, the lone psychiatrist working the night shift.[33]
Filming[edit]
Production on season one of The Night Shift began in Albuquerque, New Mexico, in late August 2013, and ended filming in mid-November.[34] Production on season two commenced on November 10, 2014.[35] Production on season three took place from February to June 2016.[36] Filming for season four began in April 2017.[37]
Reception[edit]
Critical reception[edit]
Brian Lowry of Variety said "The Night Shift is still an awfully weak blip creatively speaking".[38] David Hinckley of the New York Daily News gave the show three out of five stars.[39]
Review aggregation website Rotten Tomatoes gives Season 1 an approval rating of 20% based on 20 reviews, with a rating average of 4.3 out of 10. The site's consensus states: "Calculated and cliche-ridden, The Night Shift is DOA."[40]
At Metacritic, which assigns a weighted mean rating out of 0–100 reviews from critics, the series has a rating score of 45 based on 16 reviews, indicating "mixed or average reviews".[41]
Ratings[edit]
Season
Timeslot (ET)
Episodes
Premiered
Ended
TV Season
Rank
Viewers
(in millions)
Date
Premiere
Viewers
(in millions)
Date
Finale
Viewers
(in millions)
1
Tuesday 10:00 pm
8
May 27, 2014 (2014-05-27)
7.67[42]
July 15, 2014 (2014-07-15)
6.05[43]
2013–14
N/A
8.50[44]
2
Monday 10:00 pm
14
February 23, 2015 (2015-02-23)
5.52[45]
May 18, 2015
5.20[46]
2014–15
#86
6.67[47]
3
Wednesday 10:00 pm
13
June 1, 2016 (2016-06-01)
4.81[48]
August 31, 2016 (2016-08-31)
5.43[49]
2015–16
N/A
4.99[50]
4
Thursday 10:00 pm
10
June 22, 2017 (2017-06-22)
3.59[51]
August 31, 2017 (2017-08-31)
3.08[52]
2016–17
N/A
3.97[53]
References[edit]
Jump up ^ Rick, Porter (March 26, 2014). "'The Night Shift,' 'Food Fighters' and more NBC summer 2014 premiere dates". Zap2it. Retrieved March 27, 2014.
Jump up ^ "NBC Announces Fall Premiere Dates for New Season". The Futon Critic. June 21, 2013. Retrieved July 3, 2013.
Jump up ^ Porter, Rick (November 17, 2016). "'The Night Shift' renewed for Season 4 on NBC". TV by the Numbers. Retrieved November 17, 2016.
Jump up ^ Pedersen, Erik (March 17, 2017). "NBC Sets Summer Premieres: ‘Marlon,’ ‘Midnight, Texas,’ ‘Carmichael Show,’ Reality & More". Deadline Hollywood. Retrieved March 17, 2017.
^ Jump up to: a b Episode 1x02 "Second Chances"
^ Jump up to: a b c Episode 2x01 "Recovery"
^ Jump up to: a b c Episode 2x14 "Darkest Before Dawn"
^ Jump up to: a b c d Episode 1x01 "Pilot"
^ Jump up to: a b Episode 1x06 "Coming Home"
^ Jump up to: a b c d Episode 1x05 "Storm Watch"
Jump up ^ Andreeva, Nellie (November 22, 2016). "'The Night Shift': Ken Leung Not returning for Season 4". Deadline. Retrieved November 22, 2016.
Jump up ^ Episode 3x13 "Burned"
Jump up ^ "Jeananne Goossen (@JeananneGoossen) on Twitter". Retrieved 3 April 2016.
Jump up ^ Episode 1x03 "Hog Wild"
Jump up ^ Andreeva, Nellie (March 10, 2016). "Freddy Rodriguez To Co-Star In CBS Drama Pilot ‘Bull’". Deadline. Retrieved March 24, 2016.
Jump up ^ Episode 1x04 "Grace Under Fire"
Jump up ^ "Ask Ausiello: Spoilers on Once, Grey's, Walking Dead, Supernatural, Legends, Suits, TVD, Blindspot and More".
Jump up ^ Merle Dandridge [@MerleDandridge] (February 2, 2016). "😘Awww, LOVE! I'll be there soon!“@fromsarahjane: @MerleDandridge just not the same around @NBCNightShift without you girl! X”" (Tweet) – via Twitter.
Jump up ^ Petski, Denise (January 21, 2016). "Jennifer Beals Joins ‘The Night Shift’ As Recurring". Deadline. Retrieved January 28, 2016.
Jump up ^ Snierson, Dan (March 21, 2016). "90210 alum AnnaLynne McCord joins season 3 cast of The Night Shift". Entertainment Weekly. Retrieved March 24, 2016.
Jump up ^ "NBC’s Hit Medical Drama “The Night Shift” Returns for a Fourth Season". TV Weekly.
Jump up ^ "Night Shift Newcomer Mark Consuelos Makes Waves With a Big Save". TVLine.
Jump up ^ "Development Update: Monday, October 3". The Futon Critic. October 3, 2011. Retrieved September 12, 2013.
Jump up ^ "Development Update: Friday, August 3". The Futon Critic. August 3, 2012. Retrieved September 12, 2013.
Jump up ^ O'Connell, Michael (October 8, 2012). "NBC Orders Medical Drama Pilot from '90210' Writers". The Hollywood Reporter. Retrieved September 12, 2013.
Jump up ^ Andreeva, Nellie (April 18, 2013). "NBC Orders Backup Scripts Of ‘After Hours’". Deadline.com. Retrieved September 12, 2013.
Jump up ^ Abrams, Natalie (May 10, 2013). "NBC Orders James Spader Drama, Medical Series Night Shift and Mike O'Malley Comedy". TV Guide. Retrieved September 12, 2013.
Jump up ^ Andreeva, Nellie (October 26, 2012). "Eoin Macken To Star In NBC Pilot ‘After Hours’". Deadline.com. Retrieved September 12, 2013.
Jump up ^ "Development Update: Friday, November 2". The Futon Critic. November 2, 2012. Retrieved September 12, 2013.
Jump up ^ Rawden, Jessica (November 6, 2012). "Lost's Ken Leung And Jeananne Goossen Join NBC's After Hours Pilot". Cinema Blend. Retrieved September 12, 2013.
Jump up ^ Andreeva, Nellie (November 8, 2012). "Pilots ‘Those Who Kill’ And ‘After Hours’ Add Regulars". Deadline.com. Retrieved September 12, 2013.
Jump up ^ Roots, Kimberly (November 9, 2012). "Pilot Scoop: Royal Pains' Jill Flint Signs on to NBC Medical Drama After Hours". TVLine. Retrieved September 12, 2013.
Jump up ^ Goldberg, Lesley (July 24, 2013). "'Revolution's' Daniella Alonso Joins NBC's 'Night Shift' (Exclusive)". The Hollywood Reporter. Retrieved September 12, 2013.
Jump up ^ Gomez, Adrian (August 6, 2013). "NBC's 'Night Shift' to begin filming in ABQ". Albuquerque Journal. Retrieved September 12, 2013.
Jump up ^ "Adam Rodriguez Joins ‘The Night Shift’ As Recurring". Deadline.com. October 30, 2014. Retrieved October 30, 2014.
Jump up ^ Gomez, Adrian (February 1, 2016). "‘Night Shift’ returns for filming in Duke City". Albuquerque Journal. Retrieved February 5, 2016.
Jump up ^ @Gabestuff (4 April 2017). "And away we gooooooo......Tomorrow DAY 1 @NBCNightShift #thenightshift #nightshift" (Tweet) – via Twitter.
Jump up ^ Lowry, Brian (May 26, 2014). "TV Review: 'The Night Shift'". Variety. Retrieved May 27, 2014.
Jump up ^ Hinckley, David (May 27, 2014). "'The Night Shift': Television review". Daily News. New York. Retrieved May 27, 2014.
Jump up ^ "The Night Shift:Season 1". Rotten Tomatoes. Retrieved June 30, 2014.
Jump up ^ "The Night Shift - Season 1". Metacritic. Retrieved June 30, 2014.
Jump up ^ Bibel, Sara (May 29, 2014). "Tuesday Final Ratings: No Adjustments to 'America's Got Talent' or 'The Night Shift'". TV by the Numbers. Retrieved May 30, 2014.
Jump up ^ Kondolojy, Amanda (July 16, 2014). "Tuesday Final Ratings: 'Extreme Weight Loss' & 'Celebrity Wife Swap' Adjusted Up; 'The Night Shift' Adjusted Down + Final MLB All Star Ratings". TV by the Numbers. Retrieved July 16, 2014.
Jump up ^ Levin, Gary (September 3, 2014). "Summer 2014 Ratings". USA Today. Retrieved December 12, 2014.
Jump up ^ Kondolojy, Amanda (February 24, 2015). "Monday Final Ratings: 'The Voice' & 'Gotham' Adjusted Up; 'Castle' Adjusted Down". TV by the Numbers. Retrieved February 24, 2015.
Jump up ^ Kondolojy, Amanda (May 19, 2015). "Monday Final Ratings: 'Stalker', 'Mike & Molly' & 'Dancing With the Stars' Adjusted Up". TV by the Numbers. Retrieved May 19, 2015.
Jump up ^ de Moraes, Lisa (May 22, 2015). "Full 2014-15 Series Rankings". Deadline.com. Retrieved May 22, 2015.
Jump up ^ Porter, Rick (June 2, 2016). "Wednesday final ratings: ‘Masterchef’ premiere adjusts up, ‘Wayward Pines’ adjusts down". TV by the Numbers. Retrieved June 2, 2016.
Jump up ^ Porter, Rick (September 1, 2016). "Wednesday final ratings: 'Big Brother' adjusts up, 'America's Got Talent' and 'The Night Shift' adjust down". TV by the Numbers. Retrieved September 1, 2016.
Jump up ^ "'The Night Shift' Season Three: Ratings". TV Series Finale. June 29, 2017. Retrieved June 29, 2017.
Jump up ^ Porter, Rick (June 23, 2017). "‘Hollywood Game Night’ and ‘The Wall’ adjust up: Thursday final ratings". TV by the Numbers. Retrieved June 23, 2017.
Jump up ^ Welch, Alex (September 1, 2017). "‘Big Brother,’ ‘Zoo,’ ‘The Night Shift,’ and more adjust down: Thursday final ratings". TV by the Numbers. Retrieved September 1, 2017.
Jump up ^ "'The Night Shift' Season Four: Ratings". TV Series Finale. September 1, 2017. Retrieved September 1, 2017.
External links[edit]
Official website
The Night Shift on IMDb
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Square
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Square (disambiguation).
Square
A regular quadrilateral (tetragon)
Type
Regular polygon
Edges and vertices
4
Schläfli symbol
{4}
Coxeter diagram

Symmetry group
Dihedral (D4), order 2×4
Internal angle (degrees)
90°
Dual polygon
Self
Properties
Convex, cyclic, equilateral, isogonal, isotoxal
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles).[1] It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ {\displaystyle \square } ABCD.
Contents
[hide]
1 Characterizations
2 Properties
2.1 Perimeter and area
2.2 Other facts
3 Coordinates and equations
4 Construction
5 Symmetry
6 Squares inscribed in triangles
7 Squaring the circle
8 Non-Euclidean geometry
9 Crossed square
10 Graphs
11 See also
12 References
13 External links
Characterizations
A convex quadrilateral is a square if and only if it is any one of the following:[2][3]
a rectangle with two adjacent equal sides
a rhombus with a right vertex angle
a rhombus with all angles equal
a parallelogram with one right vertex angle and two adjacent equal sides
a quadrilateral with four equal sides and four right angles
a quadrilateral where the diagonals are equal and are the perpendicular bisectors of each other, i.e. a rhombus with equal diagonals
a convex quadrilateral with successive sides a, b, c, d whose area is A = 1 2 ( a 2 + c 2 ) = 1 2 ( b 2 + d 2 ) . {\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).} [4]:Corollary 15
Properties
A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles) and therefore has all the properties of all these shapes, namely:[5]
The diagonals of a square bisect each other and meet at 90°
The diagonals of a square bisect its angles.
Opposite sides of a square are both parallel and equal in length.
All four angles of a square are equal. (Each is 360°/4 = 90°, so every angle of a square is a right angle.)
All four sides of a square are equal.
The diagonals of a square are equal.
The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.
A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.
Perimeter and area
The area of a square is the product of the length of its sides.
The perimeter of a square whose four sides have length ℓ {\displaystyle \ell } is
P = 4 ℓ {\displaystyle P=4\ell }
and the area A is
A = ℓ 2 . {\displaystyle A=\ell ^{2}.}
In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.
The area can also be calculated using the diagonal d according to
A = d 2 2 . {\displaystyle A={\frac {d^{2}}{2}}.}
In terms of the circumradius R, the area of a square is
A = 2 R 2 ; {\displaystyle A=2R^{2};}
since the area of the circle is π R 2 , {\displaystyle \pi R^{2},} the square fills approximately 0.6366 of its circumscribed circle.
In terms of the inradius r, the area of the square is
A = 4 r 2 . {\displaystyle A=4r^{2}.}
Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:
16 A ≤ P 2 {\displaystyle 16A\leq P^{2}}
with equality if and only if the quadrilateral is a square.
Other facts
The diagonals of a square are 2 {\displaystyle \scriptstyle {\sqrt {2}}} (about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant, was the first number proven to be irrational.
A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
If a circle is circumscribed around a square, the area of the circle is π / 2 {\displaystyle \pi /2} (about 1.5708) times the area of the square.
If a circle is inscribed in the square, the area of the circle is π / 4 {\displaystyle \pi /4} (about 0.7854) times the area of the square.
A square has a larger area than any other quadrilateral with the same perimeter.[7]
A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
The square is in two families of polytopes in two dimensions: hypercube and the cross-polytope. The Schläfli symbol for the square is {4}.
The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D4.
If the inscribed circle of a square ABCD has tangency points E on AB, F on BC, G on CD, and H on DA, then for any point P on the inscribed circle,[8]
2 ( P H 2 − P E 2 ) = P D 2 − P B 2 . {\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}
If d i {\displaystyle d_{i}} is the distance from an arbitrary point in the plane to the i-th vertex of a square and R {\displaystyle R} is the circumradius of the square, then[9]
d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 3 R 4 = ( d 1 2 + d 2 2 + d 3 2 + d 4 2 4 + R 2 ) 2 . {\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}
Coordinates and equations
| x | + | y | = 2 {\displaystyle |x|+|y|=2} plotted on Cartesian coordinates.
The coordinates for the vertices of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (xi, yi) with −1 < xi < 1 and −1 < yi < 1. The equation
max ( x 2 , y 2 ) = 1 {\displaystyle \max(x^{2},y^{2})=1}
specifies the boundary of this square. This equation means "x2 or y2, whichever is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and equals 2 {\displaystyle \scriptstyle {\sqrt {2}}} . Then the circumcircle has the equation
x 2 + y 2 = 2. {\displaystyle x^{2}+y^{2}=2.}
Alternatively the equation
| x − a | + | y − b | = r . {\displaystyle \left|x-a\right|+\left|y-b\right|=r.}
can also be used to describe the boundary of a square with center coordinates (a, b) and a horizontal or vertical radius of r.
Construction
The following animations show how to construct a square using a compass and straightedge. This is possible as 4 = 22, a power of two.
Square at a given circumcircle
Square at a given side length,
right angle by using Thales' theorem
Square at a given diagonal
Symmetry
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r12 and no symmetry is labeled a1.
The square has Dih4 symmetry, order 8. There are 2 dihedral subgroups: Dih2, Dih1, and 3 cyclic subgroups: Z4, Z2, and Z1.
A square is a special case of many lower symmetry quadrilaterals:
a rectangle with two adjacent equal sides
a quadrilateral with four equal sides and four right angles
a parallelogram with one right angle and two adjacent equal sides
a rhombus with a right angle
a rhombus with all angles equal
a rhombus with equal diagonals
These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order.[10]
Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilaterals. r8 is full symmetry of the square, and a1 is no symmetry. d4, is the symmetry of a rectangle and p4, is the symmetry of a rhombus. These two forms are duals of each other and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid, and p2 is the symmetry of a kite. g2 defines the geometry of a parallelogram.
Only the g4 subgroup has no degrees of freedom but can seen as a square with directed edges.
Squares inscribed in triangles
Main article: Triangle § Squares
Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.
The fraction of the triangle's area that is filled by the square is no more than 1/2.
Squaring the circle
Squaring the circle is the problem, proposed by ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.
Non-Euclidean geometry
In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.
In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles.
In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles.
Examples:

Two squares can tile the sphere with 2 squares around each vertex and 180-degree internal angles. Each square covers an entire hemisphere and their vertices lie along a great circle. This is called a spherical square dihedron. The Schläfli symbol is {4,2}.
Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.
Squares can tile the Euclidean plane with 4 around each vertex, with each square having an internal angle of 90°. The Schläfli symbol is {4,4}.
Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72-degree internal angles. The Schläfli symbol is {4,5}. In fact, for any n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.
Crossed square
Crossed-square
A crossed square is a faceting of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih2, order 4. It has the same vertex arrangement as the square, and is vertex-transitive. It appears as two 45-45-90 triangle with a common vertex, but the geometric intersection is not considered a vertex.
A crossed square is sometimes likened to a bow tie or butterfly. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals.[11]
The interior of a crossed square can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
A square and a crossed square have the following properties in common:
Opposite sides are equal in length.
The two diagonals are equal in length.
It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
It exists in the vertex figure of a uniform star polyhedra, the tetrahemihexahedron.
Graphs
3-simplex (3D)
The K4 complete graph is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).
See also
Geometry portal
Mathematics portal
Cube
Pythagorean theorem
Square lattice
Squircle
Unit square
References
Jump up ^ W., Weisstein, Eric. "Square". mathworld.wolfram.com. Retrieved 2017-12-12.
Jump up ^ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 59, ISBN 1-59311-695-0.
Jump up ^ "Problem Set 1.3". jwilson.coe.uga.edu. Retrieved 2017-12-12.
Jump up ^ Josefsson, Martin, "Properties of equidiagonal quadrilaterals" Forum Geometricorum, 14 (2014), 129-144.
Jump up ^ "Maths is Fun - Can't Find It (404)". www.mathsisfun.com. Retrieved 2017-12-12.
Jump up ^ Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
Jump up ^ 1999, Martin Lundsgaard Hansen, thats IT (c). "Vagn Lundsgaard Hansen". www2.mat.dtu.dk. Retrieved 2017-12-12.
Jump up ^ "Geometry classes, Problem 331. Square, Point on the Inscribed Circle, Tangency Points. Math teacher Master Degree. College, SAT Prep. Elearning, Online math tutor, LMS". gogeometry.com. Retrieved 2017-12-12.
Jump up ^ Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
Jump up ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
Jump up ^ Wells, Christopher J. "Quadrilaterals". www.technologyuk.net. Retrieved 2017-12-12.
External links
Wikimedia Commons has media related to Squares (geometry).
Animated course (Construction, Circumference, Area)
Weisstein, Eric W. "Square". MathWorld.
Definition and properties of a square With interactive applet
Animated applet illustrating the area of a square
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https://af.wikipedia.org/wiki/Vierkant
Vierkant – Afrikaans
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https://an.wikipedia.org/wiki/Quadrato
Quadrato – Aragonese
an
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https://az.wikipedia.org/wiki/Kvadrat
Kvadrat – Azerbaijani
az
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https://bs.wikipedia.org/wiki/Kvadrat
Kvadrat – Bosnian
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https://br.wikipedia.org/wiki/Karrez
Karrez – Breton
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Skweya – Shona
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Kvadrat – Danish
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Quadrat – German
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interlanguage-link interwiki-de
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Kwadrat – Lower Sorbian
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https://et.wikipedia.org/wiki/Ruut
Ruut – Estonian
et
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values: Τετράγωνο – Greek

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Cuadrado – Spanish
es
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Karratu – Basque
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Cadrado – Galician
gl
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values: वर्गाकार – Hindi

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Kwadrat – Upper Sorbian
hsb
hsb
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values:
interlanguage-link interwiki-hsb
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values:
https://hr.wikipedia.org/wiki/Kvadrat
Kvadrat – Croatian
hr
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values:
interlanguage-link interwiki-hr
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values:
https://io.wikipedia.org/wiki/Quadrato
Quadrato – Ido
io
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values:
interlanguage-link interwiki-io
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Persegi – Indonesian
id
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I-square – Xhosa
xh
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values:
interlanguage-link interwiki-xh
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values:
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Ferningur – Icelandic
is
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values:
interlanguage-link interwiki-is
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values:
https://it.wikipedia.org/wiki/Quadrato
Quadrato – Italian
it
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values:
interlanguage-link interwiki-it
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values: ריבוע – Hebrew

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Pesagi – Javanese
jv
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Mraba – Swahili
sw
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Kare – Haitian Creole
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Quadrum – Latin
la
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Kvadratas – Lithuanian
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Veerkantj – Limburgish
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Efa-joro – Malagasy
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https://frr.wikipedia.org/wiki/Kwadroot
Kwadroot – Northern Frisian
frr
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values:
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Kvadrat – Norwegian
no
no
interlanguage-link-target

values:
interlanguage-link interwiki-no
display: none;

values:
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Kvadrat – Norwegian Nynorsk
nn
nn
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interlanguage-link interwiki-nn
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values:
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values:
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Kvadrat – Uzbek
uz
uz
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interlanguage-link interwiki-uz
display: none;

values:
interlanguage-link interwiki-pa
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Kwadrad – Palatine German
pfl
pfl
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values:
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values:
interlanguage-link interwiki-pms
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https://pl.wikipedia.org/wiki/Kwadrat
Kwadrat – Polish
pl
pl
interlanguage-link-target

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interlanguage-link interwiki-pl
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Quadrado – Portuguese
pt
pt
interlanguage-link-target

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Murabba – Crimean Tatar
crh
crh
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T'asra – Quechua
qu
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interlanguage-link interwiki-rue
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Square – Scots
sco
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Katrori – Albanian
sq
sq
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Quatratu – Sicilian
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Kwadrat – Silesian
szl
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Kvadrat – Serbo-Croatian
sh
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https://sv.wikipedia.org/wiki/Kvadrat
Kvadrat – Swedish
sv
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values:
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Parisukat – Tagalog
tl
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values:
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Kare – Turkish
tr
tr
interlanguage-link-target

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interlanguage-link interwiki-tr
display: none;

values:
interlanguage-link interwiki-uk
display: none;

values: مربع (ہندسہ) – Urdu

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Quadrato – Venetian
vec
vec
interlanguage-link-target

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interlanguage-link interwiki-vec
display: none;

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interlanguage-link interwiki-vi
display: none;

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Vierkant – West Flemish
vls
vls
interlanguage-link-target

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interlanguage-link interwiki-vls
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Kwadrado – Waray
war
war
interlanguage-link-target

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interlanguage-link interwiki-war
display: none;

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