The best, easiest and safest way to travel from Sugar Loaf to the bottom of Corcovado Mountain is by taxi. You could even do it by bus, but there isn’t a direct bus that connects both attractions, neither a subway station. You could also walk, but some segments of the route are deserted so walking is not recommended.
Without a doubt the best is by taxi, which will take approximately 15-20 minutes, and cost you roughly 10 to 15 US$. Once you get to the bottom of Corcovado Mountain, you can take a little train that travels up the mountain for about 20 minutes until its summit (2,300 ft above sea level), where is located the most famous Brazilian landmark: Christ the Redeemer statue.
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See photos of both here: Sugar Loaf and Christ the Redeemer statue as well as important travel information on Brazil’s most popular attractions.