# Lens Magnification

In close-up or macro photography, we want to get an image of a (usually very small) object which is as detailed and therefore as large as possible. The magnification m of a lens is defined as the relation between image size H and object size G

 m = H / G (M1)

Both are already marked in the figure used earlier for the lens equation.

By similar triangles along the central ray (yellow), this is equivalent to

 m = h / g (M2)

For a given focusing distance d, we can calculate object distance g and image distance h according to equations (F7) and (F8) as

 g = d / 2 + r (M3) h = d / 2 – r (M4)

where r is defined as

 r = sqrt (d² / 4 – f d) (M5)

With equation (M2), we can now easily calculate the magnification m of a lens as

 m = (d / 2 – r) / (d / 2 + r) (M6)

This is the formula used for the lens magnification calculator.

It is interesting to note that the magnification only depends on focal length and focusing distance, but not on other factors such as aperture or the size of the image sensor. Results are thus valid for full frame, crop and even smartphone cameras alike. However, for a given magnification, you can of course capture a larger object with a larger sensor.

Since we are interested in the maximum magnification, it is clear that we must get as close to the object as possible. Equation (F10) defines a lower bound for the minimum focusing distance (MFD) of a lens as

 d ≥ 4 f (M7)

If the bound is reached, the root term (M5) becomes zero and the magnification as derived in equation (M6) becomes 1. This is indeed the maximum magnification of most macro lenses.

## Approximation

If we solve the lens equation (L5) for h and substitute it into equation (M2), we get

 m = f / (g – f) (M8)

This formula looks much more appealing than equation (M6) and is thus sometimes published to calculate the lens magnification, but remember that we usually do not know the object distance g, but only the focusing distance d. For close-up situations, these may differ significantly.

However, if the object is reasonably far away, i.e. d is much bigger than f, we can approximate (gf) by g and g by d and get

 m ≈ f / d (M9)

This is really easy to remember. Bear in mind though that the resulting figures are much too small for macro photography.

## Notation

The magnification as derived by one of the given formulas is a positive real number. For example, for a focal length of 50 mm and a focusing distance of 0.3 m (1 ft), we get a magnification of 0.27. This means that the size of the image H is 0.27 times (or 0.27×, 27%) of the size of the original object G.

In photography, a commonly used notation for lens magnification is

 1 : M (M10)

where M is calculated as the inverse of m. In our example, the magnification is 1 : 3.7, which indicates that the size of the image is 1 / 3.7 or about 1 / 4 of the size of the original.

The next section on lens extension tubes describes how a lens can be modified to provide a higher magnification.