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How long did the space race take?
Short answer
Just under twelve years, if you date it from Sputnik 1 on 4 October 1957 to the Apollo 11 landing on 20 July 1969. Stretch the definition to the Apollo-Soyuz docking of July 1975 and it runs to nearly eighteen years. Either way, the decisive stretch was that first dozen years.

From a beeping sphere to a footprint
The starting gun is easy to place: 4 October 1957, when the Soviet Union put Sputnik 1 into orbit and a radio signal from a polished metal sphere rearranged American politics. What followed came fast. The Soviet lead held through the early years, with the first animal in orbit, the first pictures of the lunar far side and, on 12 April 1961, Yuri Gagarin as the first human in orbit. Six weeks later President Kennedy committed the United States to landing a man on the Moon before the decade was out, choosing a goal distant enough that the existing Soviet advantage would not settle it.
Twelve years, then the air went out of it
Apollo 11 landed on 20 July 1969, eleven years and nine months after Sputnik. That is the span most people mean, and it is the one that makes sense: the objective was announced publicly, both sides pursued it, and one of them arrived. The Soviet lunar programme had failed quietly in the meantime, its N1 rocket destroyed on every attempt, and the failure was not admitted for two decades. After the landing the momentum drained away on both sides, budgets fell and later Apollo flights were cancelled.
Where you stop counting
A second date has a claim. In July 1975 an Apollo command module docked with a Soyuz in Earth orbit and the two commanders shook hands through the hatch, a gesture with no engineering purpose and a very clear political one. Count to there and the race lasted nearly eighteen years, ending not with a winner but with a joint mission. Anything later belongs to a different story: the Shuttle, Mir, and eventually a space station built by both sides together with a dozen other countries.
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