Focal length for a specific distance?

This is actually a simple ratio problem: We can trace imaginary lines from the boundaries of the object being imaged to the center of the lens. A triangle is traced. The height of this triangle is the object distance = 10 meters = 10,000mm. The base of this triangle is the object’s width = 1 meter =1000mm. The ratio of height to base is 10000 ÷ 1000 =10.0

Inside the camera we can trace a similar triangle. Its height will be the focal length. Its base will be the selected dimension of the imaging sensor.

Since the object being imaged is a square 1 meter by 1 meter, lets select the sensor height which is the vertical measurement.

I think this dimension should be reduced by 3% to gain some manipulation room. Thus, let’s use 3.6 X 0.97= 3.5mm.

The focal length that meets this reasoning is discovered by multiplying the base of the image triangle by ratio of height to base of the object triangle --- thus 3.5 X 10 = 35mm.

35mm will be the zoom setting for this problem.

This works because optically we have an image triangle and an object triangle. These are similar triangles in that all angles are the same and all sides are in ratio.

Detractors will tell you that this method yields only an approximate answer because object distance is measured from the front nodal and image distance from the rear nodal. These will be separated by an unknown amount. You might improve the accuracy of this math by using the center of the lens barrel to measure object distance. I think the resulting accuracy will be good whatever.