When looking at an object such as a circle, the opposite edges of the object and your eye forms an isosceles triangle. The opposite edges of a circle are at the ends of the diameter. The circular object that appears five inches in diameter and the circular object 150 miles away form the bases of two similar isosceles triangles with the same vertex and center line. The sides and other distances of the similar triangles are proportional. For the corresponding distances of two similar triangles A and B:
base of A / base of B = side of A / side of B = center line of A / center line of B
We know the base of the apparent triangle, say, triangle A, and we know the length of the center line of the big triangle, say, triangle B, at 150 miles away, that is, the length of the center line of B is 150 miles. Given the distance at which the circle appears, the base of B, the diameter of the big triangle B, can be calculated:
base of A (center line of B / center line of A) = base of B
Let's take the distance to the center line of the apparent triangle to be 25 cm, the minimum distance at which human eyes focus. Google can tell you that 25 cm is 9.9 inches or you can convert 25 cm to inches on your own:
25 cm (1 inch / 2.54 cm) = 9.9 inches
base of B = base of A (center line of B / center line of A) = 5 inches (150 miles / 9.9 inches) = 76 miles
The base of B and the diameter of the distant circle is 76 miles across.
