In West Virginia’s White Sulphur Springs in 1926, a little Katherine Coleman is waiting and labeling the geometric figures in a stained glass window while her parents are speaking with a school official. Katherine is an accomplished student and math prodigy, so the authority wants to send her to a school for brilliant students. Since Katherine is just eight years old, she will have to miss several grades because the only school of its kind for “colored” pupils begins in the sixth grade. Katherine’s instructor offers her parents money from a collection all the teachers contributed to assist with the transfer.
A teacher at the new school invites Katherine to work on an algebra problem on the blackboard. Katherine finds the solution without hesitation and then faces her adolescent peers to explain it. The racial segregation is immediately apparent when police stop them, highlighting the challenges they face in their daily lives. After Dorothy successfully starts the vehicle, the policeman offers to drive them to the NASA Langley Research Centre with lights and sirens blazing when he gets to know about them. Watching a newsreel describing Russia’s launch of Sputnik 1, the first artificial satellite, is what the Space Task Group does. Leaders were concerned in the conversation that followed that the Soviet Union’s access to space could enable them to spy on the United States. They can’t justify the expense of a space program that doesn’t send anyone into space, so the man in charge orders AI Harrison to get them up there.
Along with many other black women employed as computers—performing math by hand—the three women are housed in the West Area Computing division, which is isolated from the rest of Langley Research Centre. Dorothy assigns the group to new tasks. Mary is assigned to help the male engineers. Dorothy is approached by her supervisor, Vivian Mitchell, for a conversation. She tells her that a new computer capable of doing analytical geometry is needed by the Space Task Group. Katherine is assigned by Dorothy because of her proficiency in analytic geometry and her general superiority in mathematics.
Since Vivian is currently doing supervisory duties, Dorothy wonders if she can advance to supervisor. However, Vivian declines, saying it's not her call. Mitchell refers to the fact that there are no intentions to appoint a "permanent supervisor for the colored group." While carrying her documents through a sealed chamber in the engineering department, Mary pauses to observe a space capsule that is due to undergo wind tunnel testing. Many of the space capsule's outer louvered heat shielding panels fly off, failing the wind test. Karl Zielinski, the chief engineer, suggests that they employ a corrugated heat shield. Mary quickly spots a design issue and proposes that they try using alternative fasteners on the panels that are already in place. Motivated by her team leader, Polish-Jewish Holocaust survivor Karl Zielinski, Mary applies to be a NASA engineer.
Katherine is escorted to the Flight Research Division on another part of campus by Vivian, who is addressed by black ladies as Mrs. Mitchell even though they know her by her first name. Vivian informs Katherine that women are expected to wear skirts that fall below the knee and to wear no jewelry other than pearls. She informs Katherine that she doesn't think Katherine will stay long because the department she's joining is demanding and has just fired several computers. She cautions her not to talk to Mr. Harrison, the department head unless he has spoken to her first. Vivian warns Katherine not to put her in public by pointing out that they have never had a person of color work in that section. Katherine walks into the enormous room filled with white males working at desks. The men mistreat Katherine, mistaking her for a caretaker.
Mr. Harrison fortunately meets Katherine and he assigns some work to do within the end of the day. In the meantime, Paul Stafford gives Katherine a tonne of work to verify. He is brief with her, assuring her that his numbers are flawless and that he needs them verified by the end of the day. Additionally, he has obscured a great deal of material, implying that Katherine needs a higher security clearance and that it is untrustworthy. When Katherine asks where the toilet is, Ruth, the only other woman in the department and a white lady, responds, "I don't know where YOUR toilet is." Urging herself outside, Katherine discovers that the building's lavatory is reserved for white women. So she needed to run half a mile out of the building to use the lavatory for colored women. She carries her work with her and keeps proofreading as she pees.
Katherine gets herself a cup of coffee, to the disapproving glare of the white members of the Space Task Group, who know that in segregated Virginia, white people and people of color don't typically drink from the same pot. Nothing is said on that day by anyone. After working as a supervisor for years, Dorothy is upset that Vivian hasn't made her a supervisor and complains about this to the other two as she drives them home that evening. Katherine has three daughters, and Donna Biscoe, Katherine's mother, looks after her kids while she works. Kids bemoan the length of time their mother has been absent; she cites her new role as the cause.
Katherine's attention is drawn to a dashing colonel while the three women attend church with their families. Following the cookout, Col. Jim Johnson approaches Katherine and strikes up a conversation. Jim makes a mistake when he finds out about Katherine’s job at NASA, stating that he is shocked that they "let women" do something "so taxing." She chastises him and walks off. On the launch pad, the NASA staff members assembled to greet the astronaut, John Glenn among them. Later on, Glenn will pilot the Friendship 7, making history as the first American in orbit. Despite being told not to, he approaches the black women standing separately from the white staff members and strikes up a conversation.
Katherine goes for coffee the following day, and she discovers a different pot marked "Colored." She also finds it empty, as no one is ready to touch or fill the coffee can. As usual, Paul hands Katherine some calculations. She couldn't answer the issues otherwise, so she had to hold it up to the light to read it. Katherine steps up and does the math, even if no one else notices. Dorothy finds out from Mitchell that NASA has deployed an IBM 7090 electronic computer that poses a danger to human-computer replacement. Al Harrison walks in and queries who fixed the equation that has been left unresolved on a big chalkboard after some time has passed.
He asks what she does, to which Katherine responds that she is double-checking her work after she admits she did. He questions how she manages to work with all the blacked-out portions when she shows him. She shows how to hold the paper up to the light and read the writing that has been blacked out. In the future, she would like uncensored reports. Al concurs, stating that there's no need to withhold information from her because she's not a Russian spy. During this discussion, Stafford is in Harrison's office and is furious.
Mitchell informs Mary that more courses are needed for the engineer role she applied for, even with her degree in mathematics and physical science.
During the ladies’ discussion, Mary added that people of color are not allowed to enroll in the only schools that offer the programs required to become engineers. Dorothy asks her to petition the court and fight for her freedom. Dorothy finds a book on FORTRAN programming when she brings her kids to the library. She's in the incorrect section, says a white woman, spotting her. Dorothy informs her that the book she desired was not available in the colored section. After being expelled from the library, she tells her kids that she brought the FORTRAN book with her as they board the bus. Dorothy claims she has the right to take the book because the library is owned by the government and she is a taxpayer, in response to her son's question.
Col. Johnson approaches Katherine at a party where many couples are dancing to music from the radio. He apologizes for treating her disrespectfully and asks her to dance. Pressure mounted to send American astronauts into space when they received news that Russian cosmonaut Yuri Gagarin had become the first man in space, completing one full orbit around the planet in 108 minutes. Katherine dazzles Harrison by solving challenging mathematical problems every day. The men faced some issues in the IBM data center. Dorothy slips into the big, unoccupied room and examines the IBM computer they've set up after noticing that a large digital computer has been installed inside.
With a few astute questions, Dorothy has also learned that her IBM, which can perform computations tenfold quicker than a room full of human computers, will replace her West Area Computing unit as soon as it demonstrates its dependability. So she learns programming for herself and her West Area coworkers by reading a book about Fortran. Against her husband's wishes, Mary petitioned to be admitted to Hampton High School, an all-white institution.
The men's issue is resolved when Dorothy, one evening in the data center, discovers that a wire on the IBM patch panel is misplaced and reconnects it properly. Katherine runs the long distance to use the loo for colored ladies on a wet day. Al Harrison visits her workstation in an attempt to find her, but she is absent once again. Harrison asks her why she leaves for so long each day when she eventually returns. Soaking wet, Katherine goes on a rant about how there are no restrooms she is permitted to use in the building they are working in; therefore, she must run half a mile to the west campus to use the colored women's restroom.
She continues to say that she's never been permitted to wear any jewelry other than pearls. And NASA doesn't pay colored computers enough to buy pearls, nor is she allowed to sip coffee from the same pot as everyone else. Al pays close attention. He stays silent, but after Katherine storms out, he goes to the coffee station and tears the label labeled "Coloured" off of the coffee maker. In the following scene, Al uses a crowbar to attack the colored women's toilet sign at the West Area Computing division. He declares that there won't be any toilet segregation once it falls, saying, "Here at NASA, we all pee the same color."
Over the ensuing months, Jim visits Katherine and her family frequently. Mary approaches a judge to request permission to enroll in an all-white institution to complete her engineering degree. She convinces the local judge to let her attend night school by pleading her case in court and appealing to his historical sensibilities. Assigned to produce reports, Katherine places her name on the byline after Paul Stafford's because she did most of the research. Following her request, Al permits Katherine to participate in high-level discussions aimed at determining the space capsule's point of re-entry. Asserting that "computers" cannot write reports, Stafford forces Katherine to erase her name from them and gives Stafford all credit for her work.
Whenever Katherine attends the discussion, Paul gets pissed off and refuses to take Katherine to the meetings where fresh data is being discussed. Suddenly, one day, the male technicians barge in and ask Dorothy what she's doing. She was testing a program on a set of punch cards. When they discover that Dorothy, who has been tinkering around in her free time, has accomplished something that has evaded them for weeks—getting the computer to run a program and get useful results—they are relieved that she won't break anything.
Katherine and Paul Stafford are at odds over credit for the reports they both contribute to. The details of the friendship mission are changing so quickly Katherine's calculations are frequently incorrect by the time she completes them. Katherine contends that she must attend meetings where fresh knowledge is discussed, but Paul dismissively states that women are not permitted. She informs Al Harrison that she would want to attend the meetings because she is in charge of revising the launch calculations any time there is a modification of any type, including changes to the landing area, payload, launch time, etc. Changes are usually discussed at these sessions. When Al mentions that women are not expected to attend meetings, she counters that a guy circling the Earth is also not expected to follow any protocol. As Katherine notes, he is the boss and sets the rules; all he needs to do is act like one.
Al grants her permission to observe despite Paul's objection. But he orders her to remain silent. Al leads Katherine into the meeting with John Glenn and other prominent NASA figures. They need a technique to determine Glenn's Friendship 7 space capsule's landing location; one gripes to Al. Al is silent for a moment, thinking over the issue. Subsequently, he extends a piece of chalk towards Katherine and requests that she attempt it.
Though unprepared, Katherine considers her options and covers half of the whiteboard with a series of equations that indicate Glenn will splash down at a specific latitude and longitude close to the Bahamas based on the present plans. Everyone is impressed—including John Glenn—as they are unable to find any weaknesses in her reasoning or calculations.
Al has threatened to withhold payment from the men who are struggling to get the machine to operate. Dorothy explains that she has mastered FORTRAN and has been instructing the computers in her group about FORTRAN and the IBM machine when Vivian observes that they are lacking computer programmers. Dorothy makes plans for all thirty of the ladies under her supervision to accompany her and become data center employees. As they all go together to their new section, Dorothy informs the women that they have all been reassigned.
Mary becomes the first woman of color to attend classes when she is allowed to enroll only at night. Katherine is still calculating trajectories and producing reports as John Glenn's launch draws near, but she can now put her name in the bylines because Paul has now moved past his racist grievance and she is greatly valued by the project. In the lavatory Vivian encounters Dorothy. She expresses regret to Dorothy for never assigning her to be a supervisor. Dorothy is informed by Vivian that she has nothing against the black ladies who work for her and that she has always done what is best for them. Dorothy gives her a quick look before leaving and cynically remarks, "I know you believe that."
When Katherine returns home one evening, her girls are dressed for a special occasion, and her dinner table is prepared accordingly. Having prepared them dinner, Jim Johnson exits the kitchen carrying a plate of food and places a small jeweler's box on the table in front of Katherine. He follows up with heartfelt remarks about how he's joining the family and not just marrying Katherine, and she accepts his proposal.
Al, however, informs her that the Space Task Group no longer needs Katherine because Friendship 7 is set to launch. Ruth gives Katherine a wedding present from the entire department, a strand of pearls, as she packs up her belongings, moves back to the West Area, and marries Jim. Everybody watches the launch of John Glenn on television. When Glenn's flight is calculated by the IBM computer differently than the day before, there is a problem in the control room.
It can only be safe to take off with precise figures; thus, one set of calculations must be accurate. Because Glenn has more faith in Katherine's intellect than in the IBM computer, Glenn asks that she complete the math by hand. Al sends Sam to find her, and he heads to the West Area Computing room to check the calculations at her former desk. After running the calculations and determining which set of data is correct, Katherine dashes back to the control room to give Al the notebook. Katherine is faced with a closed door. She's rescued the day, but she stands outside, sad. After a long moment, Al comes back and gives her a pass to enter the control room, escorting her inside alongside him.
Katherine and Al watch from the control room as fifty million people watch the lift-off on television. As John Glenn maneuvers Friendship 7 into orbit, we can glimpse the horizon from within the spacecraft. At last, Vivian addresses Dorothy as Mrs. Vaughan and elevates her to the position of supervisor in the Analysis and Computation Division. Following successful orbits of Earth, Glenn detects a warning signal indicating a problem with the heat shield fasteners.
Glenn, who is experiencing a rough journey and quickly notices that the capsule is getting quite hot, is given this advice. She informs them that instead of throwing the retro pack away as intended for re-entry, Glenn needs to leave it in place since the straps could hold the heat shield in place. Glenn will perish in the capsule's burning as it re-enters the atmosphere if the heat shield blows apart. After a nervous few minutes, the control room loses touch with Friendship 7. Glenn safely splashes down because the straps hold down enough of the heat shield to save his life. John Glenn completed three of a scheduled seven-orbit flight. His Friendship 7 mission turned the tide in the space race, propelling NASA to the moon in 1969. Mary earned her engineering degree and went on to become NASA's first female African American engineer; Dorothy went on to become NASA's first African American supervisor;
Katherine continued by calculating the trajectories for both Apollo 13 and the 1969 Apollo 11 lunar mission. In 2015, she received the Presidential Medal of Freedom from President Obama. She worked at the Virginian Langley Research Centre, which was renamed the Katherine G. Johnson Computational Research Facility in 2016. She married Jim Johnson in 1986, and they are still together today.
