What is the easiest way to measure the distance between the earth and the sun?
The easiest way to calculate the distance from the earth to the sun is by using the known distance to one of the other planets. It takes a few steps though.. so bear with me. Take Venus for example; we can use radar signals to find the distance to Venus. Signals sent from earth will bounce off the surface of Venus and then travel back to earth where we can detect them. Because radar travels at the speed of light (a quantity that is well known from laboratory experiments) you can easily calculate the distance between Venus and Earth at any time.
distance = 0.5*(time for radar to travel both ways)*(speed of light)
Now that we know the distance to Venus we can calculate the distance from the Earth to the Sun using trigonometry. Both Venus and Earth travel in almost perfect circles around the earth, so this is a fairly easy calculation. It might help to envision the situation if you draw two concentric circles on a piece of paper to represent the orbits of the Venus and Earth. Now imagine that the earth is set at one position in its orbit, as Venus travels along its orbit there are two points where the Venus-Sun line and the Earth-Sun are perpendicular (90 degrees apart). From our perspective on earth this is the point where Venus is the farthest it will get from the Sun. By keeping track of Venus over time we determine exactly when this will happen. When this happens we make the radar measurements mentioned above and we also measure the angle between the Sun and Venus at this point (here's where the trigonometry comes in!). It turns out that the distance from the Sun to the Earth is:
(distance of Sun-Earth) = (distance of Earth-Venus)*(cosine(angle))
This is how we get the distance of Sun-Earth to be 92,955,830 miles! This is also known as one astronomical unit (AU)