I'm hoping to prove this distance to camera equation:

Question

$$ D = \frac{f\times H}{h} $$ where

$$ \begin{align} D &= \textrm{distance from lens to object} \\ (d &= \textrm{distance from lens to 35mm film}) \\ f &= \textrm{focal length} \\ H &= \textrm{height of object} \\ h &= \textrm{height of object's image on 35mm film} \end{align} $$

I've seen some form of this equation in several places including these two threads:

• How do I calculate the distance of an object in a photo?
• Is the formula for object image size given focal length, etc. independent of sensor size?

The problem I'm having is with \$f\$. From the magnification for a thin lens:

$$ M = \frac{D}{d} = \frac{H}{h} = \frac{f}{f-D} $$

I can't figure out how to arrive at \$D = f\cdot H/h\$.

Answer 1

An object 1 meter (1000mm) in height is photographed with a 50mm focal length lens. The object distance is 5meter (5000mm).

H = actual height = 1000

D = actual distance = 5000

We can trace out a triangle with the vertex at the lens. The base of this triangle is the objects height; the height of this triangle is object distance. The ratio of height to distance is 1000 ÷ 5000 = 0.200.

Inside the camera, we can trace a similar triangle. A similar triangle means the angles of both triangles have identical angles and the ratio of corresponding sides will be identical.

The height of this image triangle is the focal length of the lens d = 50

The base of this triangle is unknown. We can calculate; 50 X 0.2 = 10

Thus h = 10mm

Prove formula D = f(H/h)

Solve:

f=50

H=1000

h = 10

D = 50 X (1000/10)

D =50 X 100

D = 5000 (actual object distance = 5 meters)

Answer 2

What you're trying to arrive at can't be right. Since \$D/d = H/h\$ $$ D = \frac{fH}{h} $$ is equivalent to $$ D = \frac{fD}{d} $$ Dividing both sides by \$D\$: $$ \begin{align} 1 &= \frac{f}{d} \\ d &= f \end{align} $$ Which clearly isn't correct.