In macro photography, we want to get an image of a usually very small subject which is as detailed and therefore as large as possible. The *magnification* **m** is defined as the ratio of the image size **H** (on the sensor) to the subject size **G** (in the real world)

m | = | H / G | (M1) |

The situation is illustrated in the following figure, which was already introduced for the lens equation.

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

m | = | h / g | (M2) |

For a given focus distance **d**, subject distance **g** and image distance **h** are given by equations (F7) and (F8), respectively, 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** as

m | = | (d / 2 – r) / (d / 2 + r) | (M6) |

This is the lens magnification equation used for the lens magnification calculator.

It is interesting to note that the magnification only depends on focal length **f** and focus distance **d**, 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 subject with a larger sensor.

Since we are interested in the maximum magnification, it is clear that we must get as close to the subject as possible. Equation (F10) defines a lower bound for the minimum focus 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.

## 5.1 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 subject distance **g**, but only the focus distance **d**. For close-up situations, these may differ significantly.

However, if the subject is reasonably far away, i.e. the focal length **f** of the lens becomes insignificant compared to the focus distance **d**, we can approximate (**g** – **f**) by **d** and get

m | ≈ | f / d | (M9) |

Double the focal length and you double the magnification. Easy to remember. Bear in mind though that the resulting figures are much too small for macro photography.

## 5.2 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 focus 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 subject **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.