A burning question

Remember Aristarchus of Samos? He was an early Greek astronomer who suggested that the Earth orbited around the Sun (a heliocentric model). He also answered a fascinating question: how far away is the Sun?
This Extreme Ultraviolet Imaging Telescope (EIT) image of a huge, handle-shaped prominence was taken on Sept. 14,1999. Credit: NASA/European Space Agency
He realized that we could figure out the distance to the sun relative to the distance to the moon. He noted that the sun, moon, and earth form a right triangle (right angle at the moon) during the first and last quarter moons.
Image Credit: NASA Goddard Space Flight Center
A quarter moon occurs when the moon appears to be half illuminated from our vantage point. When the moon is seen to be exactly in the first quarter phase the sun-moon-earth angle is exactly 90 degrees. This means a line drawn from Aristarchus’ position to the moon and from the moon to the Sun formed a right angle.
Image is not to scale. Image Credit: Brit Cruise
Aristarchus simply needed to calculate the angle x. That is equal to how far he would have to tilt his telescope between the moon and sun (if telescopes had been invented yet). He calculated this angle as 87 degrees, though the true value is closer to 89.83 degrees (a difficult measurement to get right with the naked eye!).
We already know the distance to the moon is 384,400 km. Finally we have all the information needed! We apply the cosine function to find the hypotenuse of the triangle which is the distance to sun:
Image not to scale

cos(angle) = adjacent / hypotenuse

cos(89.83) = moon distance / sun distance

sun distance = 384,400 / cos(89.83)

sun distance = 129,556,058 km

This is estimate off by 13.3% which is quite good for a rough approximation. Compare this to the actual value:

Average distance to the Sun: 149,600,000 km

This is the basis for the astronomical unit (abbreviated as AU) which is based on the Earth-Sun distance.

1 AU = 149 597 871 km

Challenge question: How could we measure the size of the sun using this new information?
hint: we need to review similar triangles. Let's do that now!