How old is the planet Earth?
Short answer: About 4.6 billion years old.
Long answer: The Sun, the planets and almost everything else in our Solar System were all formed together from a spinning cloud of dust and gas1. Our current best estimate is that this happened around 4.6 billion years ago, which means that Earth is that old.
It's not easy to find direct evidence about the age of our planet, because it has changed so much over time. There's nothing left of the original surface. However, scientists can piece together the evidence we do have to form a fairly good picture.
Evidence found on Earth includes sedimentary rock layers, fossils, radiometric dating and more.
Evidence found in space includes samples from the Moon, asteroids and comets.
In addition to the physical evidence, we have a theoretical framework to explain how and when the Solar System was formed. It's not perfect or complete but it does convincingly match what we see in all the objects of the Solar System. We can also observe other Solar Systems and see them in various stages of development, and we find that they all match our theory too. Despite some gaps and problems in our understanding, we're pretty confident that we have a good estimate of the age of our Solar System.
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Author: Dave Owen
Footnotes
1. According to the "Nebular hypothesis". See en.wikipedia.org/wiki/Nebular_hypothesis