Question

Suppose Eratosthenes had found that, in Alexandria, at noon on the first day of summer, the line to the Sun makes an angle 30° with the vertical. What, then, would he have found for Earth’s circumference?

Answer 1

If Eratosthenes observed a 30° angle at Alexandria, he would have calculated Earth's circumference to be 30,000 stadia using the distance between Alexandria and Syene and the proportion associated with the angle.

Step-by-step explanation:

Suppose Eratosthenes had discovered that the angle of the Sun's rays to the vertical in Alexandria at noon on the first day of summer was 30° instead of the historical 7.2°, he would have calculated a different value for the Earth's circumference. Using his method, the observed angle corresponds to the arc between Syene and Alexandria. If the angle observed was 30°, that would correspond to ⅓ of a complete 360° circle. Since Syene and Alexandria were 5000 stadia apart, knowing this measurement equates to ⅓ of the Earth's circumference, Eratosthenes would have calculated the Earth's total circumference to be 5000 stadia × ⅝⅓, which simplifies to 5000 × 6, or 30,000 stadia.

Answer 2

6000 miles

Step-by-step explanation:

The angle of the sun shone at an angle of 30° to the zenith

This means that the angle of the sector of the circle is 30° (θ)

S = Length of the sector of the circle = 500 miles (Distance between Alexandria and Syene)

r = radius of earth

Converting 30° to radians

`\theta=30*(\pi)/(180)`

`S=r\theta\\\Rightarrow r=(S)/(\theta)\\\Rightarrow r=(500)/(30*(\pi)/(180))\\\Rightarrow r=954.93\ Miles`

Circumference

`C=2\pi r\\\Rightarrow C=2\pi (500)/(30*(\pi)/(180))\\\Rightarrow C=6000\ Miles`

∴ Earth’s circumference would have been 6000 miles