How do you measure the distance between Earth and the Sun?

How do we calculate the distance of the Sun from the Earth, or the actual size of the Sun, or the speed of travel of Earth in its orbit around the Sun? Clearly, from an answer to one of these questions one can find out the answers to the others. But how do we find the first answer?

It’s pretty amazing how well we know the distance between Earth and Moon so accurately

The first step in measuring the distance between the Earth and the Sun is to measure the distance between Earth and another planet in terms of the distance between Earth and the Sun. So, let us assume that the distance between Earth and the Sun is "a". Now, let us consider the orbit of Venus. To first approximation, the orbits of Earth and Venus are perfect circles around the Sun.

Take a look at the diagram below (not to scale). From the representation of the orbit of Venus, it is clear that there are two places where the Sun-Venus-Earth angle is 90 degrees. At these points, the line joining Earth and Venus will be a tangent to the orbit of Venus. These two points indicate the greatest elongation of Venus and is the farthest that Venus will get away from the Sun in the sky.

Another way to understand this is to look at the motion of Venus in the sky relative to the Sun: as Venus orbits the Sun, it gets further away from the Sun in the sky, reaches a maximum separation from the Sun (corresponding to the greatest elongation) and then starts going towards the Sun again. This by the way is the reason why Venus is never visible in the evening sky for more than about three hours after sunset and in the morning sky more than 3 hours before sunrise. 

Now, by making observations of Venus in the sky, one can determine the point of greatest elongation. One can also measure the angle between the Sun and Venus in the sky at the point of greatest elongation. In the diagram, this angle will be the Sun-Earth-Venus angle marked as "e" in the right angled triangle. Now, using the trigonometry, one can determine the distance between Earth and Venus in terms of the Earth-Sun distance:

distance between Earth and Venus = a * cosine(e).

Now, the distance to Venus can be measured by radar measurements, where a radio wave is transmitted from Earth and is received when it bounces off Venus and comes back to Earth. By measuring the time taken for the pulse to come back, the distance can be calculated as radio waves travel at the speed of light. Once this is known, the distance between Earth and Sun can be calculated.

Historically, the first person to do this measurement was Aristarchus (310-230 BC). He measured the angular separation of the Sun and the Moon when its phase was first or third quarter to derive the distance between Earth and Sun in terms of the distance between the Earth and the Moon.

Eratosthenes (276-194 BC) also measured the distance between Earth and Sun as 804,000,000 stadia. The first scientific measurement of the Earth-Sun distance was made by Cassini in 1672 by parallax measurements of Mars (he observed Mars from two places simultaneously).

As you have indicated, once the distance between Earth and Sun is known, one can calculate all the other parameters. We know that the Sun subtends an angle of 0.5 degrees. Again, using trigonometry, the radius/diameter of the Sun can be calculated from the distance between Earth and Sun, d, as R_sun = tan(0.5 degrees) * d. Also, since we know the time taken by the Earth to go once around the Sun (P = 1 year), and the distance traveled by the Earth in this process (2*pi*a), we can calculate the average orbital speed of Earth as v = P/(2*pi*a).

Anyway, the relevant numbers are:

Earth-Sun distance, d = roughly 150 million km (defined as 1 Astronomical Unit)
 

Radius of the Sun, R_sun = roughly 700,000 km  

Orbital speed of Earth, v = roughly 30 km/s

References:

1. Great Moments in the History of Solar Physics 1.
2. Eratosthenes of Cyrene.
3. Historical attempts to detect Astronomical parallax

4. Astro Cornell