Katherine Goble works at the West Area of Langley Research Center in Hampton, Virginia, in 1961, alongside her colleagues Mary Jackson and Dorothy Vaughan, as lowly "computers", performing math calculations without being told what they are for. All of them are African-American women; the unit is segregated by race and sex. White supervisor Vivian Mitchell assigns Katherine to assist Al Harrison's Space Task Group, given her skills in analytic geometry. She becomes the first Black woman on the team; head engineer Paul Stafford is especially dismissive.
Mary is assigned to the space capsule heat shield team, where she immediately identifies a design flaw. Encouraged by her team leader Karl Zielinski, a Polish-Jewish Holocaust survivor, Mary applies for a NASA engineer position. She is told by Mitchell that, regardless of her mathematics and physical science degree, the position requires additional courses. Mary files a petition for permission to attend all-white Hampton High School, despite her husband's opposition. Pleading her case in court, she wins over the local judge by appealing to his sense of history, allowing her to attend night classes.
Katherine meets African-American National Guard Lt. Col. Jim Johnson, who voices skepticism about women's mathematical abilities. He later apologizes and begins spending time with Katherine and her three daughters. The Mercury 7 astronauts visit Langley, and astronaut John Glenn goes out of his way to greet the West Area women. Katherine impresses Harrison by solving a complex mathematical equation from redacted documents, as the Soviet Union's successful launch of Yuri Gagarin increases pressure to send American astronauts into space.
Harrison confronts Katherine about her "breaks," unaware that she is forced to walk half a mile (800 meters) to use the nearest restroom designated to "colored" people. She angrily explains the discrimination she faces at work, which leads Harrison to destroy the "colored" restroom signs and abolish restroom segregation. He allows Katherine to be included in high-level meetings to calculate the space capsule's re-entry point. Stafford makes Katherine remove her name from reports, insisting that "computers" cannot author them, and her work is credited solely to Stafford.
Informed by Mitchell that there are no plans to assign a "permanent supervisor for the colored group," Dorothy learns NASA has installed an IBM 7090 electronic computer that threatens to replace human computers. When a librarian scolds her for visiting the whites-only section, Dorothy sneaks out a book about Fortran and teaches herself and her West Area co-workers programming. She visits the computer room, successfully starts the machine, and is promoted to supervise the Programming Department; she agrees to do so if thirty of her co-workers are transferred as well. Mitchell finally addresses her as "Mrs. Vaughan".
Making final arrangements for John Glenn's launch, the department no longer needs human computers; Katherine is reassigned to the West Area and marries Jim. On the day of the launch, discrepancies are found in the IBM 7090 calculations, and Katherine is asked to check the capsule's landing coordinates. She delivers the results to the control room, and Harrison allows her inside. After a successful launch and orbit, a warning indicates the capsule's heat shield may be loose. Mission Control decides to land Glenn after three orbits instead of seven, and Katherine supports Harrison's suggestion to leave the retro-rocket attached to help keep the heat shield in place. Friendship 7 lands successfully.
Though the mathematicians are ultimately replaced by electronic computers, a textual epilogue reveals Mary obtained her engineering degree and became NASA's first female African American engineer; Dorothy continued as NASA's first African American supervisor; and Katherine, accepted by Stafford as a report coauthor, went on to calculate the trajectories for the Apollo 11 and Space Shuttle missions. In 2015, she was awarded the Presidential Medal of Freedom. In 2016, NASA dedicated the Langley Research Center's Katherine Johnson Computational Building in her honor.
