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How far away is the Moon? Hipparchus first calculated the distance in 190BC using a simple method involving trigonometry and was accurate to 1,000km!

In 190 BC, Hipparchus calculated the distance to the moon as 397,000km using simple trigonometry. Modern laser guided measurements have shown that the average distance to the Moon is 382,000km. So how did Hipparchus achieve this remarkably accurate measurement over 2,000 years ago?

Hipparchus wasn't the first to try and calculate this distance. Both Eratosthenes and Aristarchus tried, but it was Hipparchus who was able to satisfy all his peers that his assumptions and hypothesis were accurate.

In order to calculate the distance to the Moon, we must first consider two observers on the surface of Earth.

The first observer, A, can see the Moon on the horizon, while the second observer, B, sees the Moon directly overhead at exactly the same time.

Calculate Distance to the Moon

Calculate Distance to the Moon

The distance between the two observers can be used to calculate the angle, theta θ, and using some basic trigonometry we can solve for d.

Moon distance calculation
Equation 4 - Moon distance calculation

Where a is the angular distance between the two observers and r is the radius of the Earth. The radius of the Earth had already been calculated nearly one hundred years earlier by Eratosthenes so we just need to measure the distance between the two observers to find the angle tan θ. We can then calculate d using the formula below.

Tan theta solved for d
Equation 5 - Tan theta solved for d

Using this exact method, Hipparchus was able to calculate the distance as 59 Earth radii, or 397,000km. This is very close to the modern measured figure of 382,000km.

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