The focal length is distance lens to sensor (film) when imaging an object at infinity (as far as the eye can see ∞). This measurement will be elongated if the object is closer than infinity.
Suppose you photograph flagpole 10 feet tall from a distance of 20 feet using a camera with 50mm lens. How tall will the image of flagpole be on film or digital sensor?
Ratio to the rescue:
10 ÷ 20 = 0.5. This is the ratio of height to distance. A ratio is unique as it is dimensionless. With a 50mm mounted the image forms about 50mm downstream. The image height to focal length will have the same ratio. We multiply the focal length by the ratio to obtain the image height on film or sensor. Thus 50 x 0.5 = 25mm is the flagpole’s image height.
Now your desire is to image a 1 inch object and obtain a life-size image. We call this magnification1 or “unity” or an object to image ratio of 1:1.
For any lens, regardless of focal length, when we image at unity, the distance between lens and object is always twice the focal length. Thus for a 50mm lens, the object placement is 100mm forward. The image distance (back focus) is also twice the focal length. Thus to focus the lens is racked forward 100mm away from film or sensor. Many cameras do not allow this much forward movement of the lens during focusing. In this case we impose a spacer ring between lens and camera body.
For unity (life-size 1:1) the distance object to film and sensor is 4X the focal length. Thus the object to film/sensor distance is 4x50 = 200mm. This is useful knowledge because the measuring point of the lens (rear nodal position) is not likely at the center of the lens barrel. Knowing that to image at unity, the distance object to sensor (film) is 4X the focal length is useful for correct object placement.
One could use the published angle of view and use trigonometry to do these calculations. However, the published angle of view is often confusing. As an example: A 50mn lens mounted on a full frame camera has an image size (format) measuring 24mm height by 36mm length. The diagonal measure of this rectangle is 43.3mm. The angle of view most often published is 46⁰. This is the diagonal angle of view. I think this is useless data. The two useful numbers are 27⁰ vertical and 40⁰ horizontal. The lens maker prefers to publish the bigger value. This is like TV’s being sized by their diagonal measure.