How Is Magnification Calculated? Formula & Examples

This calculator works out the magnification of an optical system (such as a microscope, telescope, or magnifying glass) from the focal lengths of the objective and eyepiece, or from the object and image distances. You enter the relevant focal lengths or distances, and you get the magnification factor, plus the total magnification for multi-lens systems. The basic formula is M = f_objective / f_eyepiece for telescopes, and M = (image distance) / (object distance) for simple lenses.

Formula

For a telescope or microscope with an objective lens (or mirror) of focal length f_obj and an eyepiece of focal length f_eye, the magnification is:

M = f_obj / f_eye

where M is the magnification (written as ×, e.g., 50×), f_obj is the focal length of the objective in millimeters (mm), and f_eye is the focal length of the eyepiece in millimeters (mm). The units cancel, so magnification has no unit.

For a single thin lens (like a magnifying glass) where the object is at a distance d_o and the image forms at distance d_i, the magnification is:

M = d_i / d_o

Here d_i and d_o are in the same unit (both mm, cm, etc.).

Worked Example

Using the default values from the calculator: a telescope with an objective focal length of 100 mm and an eyepiece focal length of 10 mm.

1. Write the formula: M = f_obj / f_eye.
2. Plug in the numbers: M = 100 mm / 10 mm.
3. Divide: M = 10.
4. The magnification is 10×.

If you add a 2× Barlow lens (which doubles the effective focal length of the objective), the total magnification becomes 10 × 2 = 20×, as shown in the calculator.

When to Use Each Formula

• Telescope or microscope with two lenses: Use M = f_obj / f_eye. This applies to refractors, reflectors (using the mirror’s focal length), and compound microscopes.
• Simple magnifier (one lens): Use M = d_i / d_o. This is for a magnifying glass or a single lens used to view a nearby object.
• Camera or projector: The same M = d_i / d_o applies, where d_i is the distance from lens to sensor/film and d_o is the distance from lens to the subject.

Common Mistakes and Misconceptions

• Using the wrong focal length: For a telescope, the objective focal length is the distance from the primary lens/mirror to where it focuses light, not the length of the tube. For a microscope, it’s the tube length divided by the objective’s magnification (often 160 mm / 40 = 4 mm).
• Mixing units: Both focal lengths must be in the same unit. If one is in mm and the other in cm, convert first.
• Confusing magnification with resolution: Magnification makes things appear larger, but resolution is the ability to see fine detail. A high magnification with poor resolution just gives a blurry big image.
• Thinking higher magnification is always better: Beyond a certain point (about 2× the aperture in mm for telescopes), extra magnification shows a dimmer, blurrier image. The useful magnification is limited by the aperture.

How to Choose the Right Eyepiece

To get a desired magnification, rearrange the formula: f_eye = f_obj / M. For example, with a 1000 mm telescope and you want 100×, you need a 10 mm eyepiece. For 50×, a 20 mm eyepiece. Most telescopes come with one or two eyepieces, but you can buy additional ones to get different magnifications. A good rule for telescopes: the maximum useful magnification is roughly 2× the aperture in millimeters (or 50× per inch of aperture). For a 100 mm telescope, that’s about 200×.

Limits of the Formula

The simple formulas assume thin lenses and paraxial rays (rays close to the optical axis). Real lenses have aberrations, and thick lenses or multi-element eyepieces have effective focal lengths that may differ from the nominal value. Also, for a magnifying glass, the magnification depends on how close you hold it to your eye and the object. The formula M = d_i / d_o gives the lateral magnification, but the angular magnification (what you perceive) is often given as M = 25 cm / f for a magnifying glass, where 25 cm is the near point of a typical eye. That’s a different concept – it compares the angle subtended by the image to the angle subtended by the object at the near point.

Frequently Asked Questions

What does 10× magnification mean?

It means the object appears 10 times larger in linear dimension (width or height) than it does to the naked eye at the same viewing distance. For example, a 1 mm letter appears as if it were 10 mm wide. The area appears 100 times larger (since area scales as the square of linear magnification).

How do I calculate magnification for a microscope?

For a compound microscope, multiply the objective magnification by the eyepiece magnification. For example, a 40× objective with a 10× eyepiece gives 400× total. The objective magnification is usually printed on the side (e.g., 4×, 10×, 40×, 100×).

What is the difference between magnification and resolution?

Magnification is how much larger an object appears. Resolution is the ability to distinguish two closely spaced points as separate. You can magnify a blurry image, but you can’t resolve details that are beyond the optical system’s resolution limit. For telescopes, resolution is determined by the aperture; for microscopes, by the numerical aperture and wavelength of light.

Can I use the same formula for a camera zoom lens?

No. A camera zoom lens is described by its focal length range (e.g., 18–55 mm), but the magnification of the image on the sensor depends on the focus distance. The formula M = d_i / d_o applies, but d_i changes as you focus. For distant objects, d_o is very large, so M is very small (much less than 1).

Why does my telescope’s magnification seem lower than the formula says?

Possible reasons: the eyepiece focal length is not what you think (some cheap eyepieces have a different effective focal length), the telescope’s actual focal length is shorter than the advertised value (due to mirrors or lenses), or you are viewing a very distant object and the atmosphere limits the sharpness. Also, if you are using a Barlow lens or a diagonal, it may affect the effective focal length.

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in f_obj | Objective focal length | mm | 100 | 0
in f_eye | Eyepiece focal length | mm | 10 | 0
out mag | Magnification | × | f_obj / f_eye | 1
out total | Total magnification (if 2x Barlow) | × | (f_obj / f_eye) * 2 | 1