Calculate Magnification from Focal Length

This calculator works out the magnification of a simple optical system (like a telescope, binoculars, or a camera with a fixed lens) when you enter the focal length of the objective (the main lens or mirror) and the focal length of the eyepiece (or the effective focal length of the camera sensor system). You get the linear magnification as a ratio (e.g., 50×) and the same value expressed as a decimal. The formula is straightforward: magnification = objective focal length ÷ eyepiece focal length.

Formula

For a telescope, binoculars, or a simple magnifier, the angular magnification M is given by:

M = f_objective / f_eyepiece

Where:

• f_objective = focal length of the objective lens or mirror (in millimeters, cm, or any unit, as long as both are the same)
• f_eyepiece = focal length of the eyepiece (in the same unit)

The result is a dimensionless number. A magnification of 10 means the object appears 10 times larger in angular size than with the naked eye. For a camera, the same formula applies if you replace the eyepiece with the sensor’s effective focal length (often the lens focal length for a simple fixed-lens camera).

Worked Example

Using the calculator defaults: an objective focal length of 500 mm and an eyepiece focal length of 25 mm.

1. Write the formula: M = 500 / 25
2. Divide: 500 ÷ 25 = 20
3. So the magnification is 20×.

The calculator shows 20.0× and 20.0:1, which matches.

Understanding Focal Length and Magnification

Focal length is the distance from the lens (or mirror) to the point where parallel light rays converge to a focus. In a telescope, the objective has a long focal length, and the eyepiece has a short one. The ratio of those two distances gives the magnification. A longer objective focal length or a shorter eyepiece focal length both increase magnification. However, magnification is not the only thing that matters—aperture (the diameter of the objective) determines how much light is gathered, and too much magnification can make the image dim or blurry.

For binoculars, the magnification is usually printed as something like “10×42” where 10 is the magnification and 42 is the objective diameter in millimeters. You can verify this by dividing the objective focal length (which is not the same as the objective diameter) by the eyepiece focal length, but manufacturers often don’t list those numbers separately.

How to Choose the Right Eyepiece

If you have a telescope with a known focal length (say 1000 mm) and you want a specific magnification, you can rearrange the formula to find the eyepiece focal length you need:

f_{eyepiece = fobjective / M}

For example, to get 50× with a 1000 mm telescope, you need a 20 mm eyepiece (1000/50 = 20). Common eyepiece focal lengths are 25 mm, 20 mm, 10 mm, and 6 mm. A good rule of thumb is that the maximum useful magnification is about 2× per millimeter of aperture (e.g., a 100 mm telescope can handle up to 200× on a clear night). Beyond that, the image gets too dim or distorted.

Common Mistakes and Misconceptions

One frequent error is mixing units. If the objective focal length is in meters and the eyepiece is in millimeters, you must convert both to the same unit. Another mistake is confusing magnification with zoom. Zoom implies a variable focal length, while magnification is a fixed ratio. Also, some people think that higher magnification is always better—but at high magnification, the field of view narrows, the image dims, and atmospheric turbulence becomes more visible. A telescope with a small aperture and high magnification will show a dim, fuzzy image.

Another misconception: for a magnifying glass, the formula is different. A simple magnifier uses the near point of the eye (usually 25 cm) divided by the focal length of the lens. That formula only applies to a single lens held close to the eye, not to a two-lens system like a telescope.

Limits of the Formula

The simple ratio formula works for afocal systems (where the final image is at infinity, like a telescope focused for your eye). It does not account for the eye’s own focal length or the distance from the eyepiece to your eye. For a camera with a zoom lens, the magnification at the sensor is given by the lens focal length divided by the sensor’s diagonal (or width), but that’s a different concept—it’s the reproduction ratio, not angular magnification. For macro photography, magnification is the ratio of the image size on the sensor to the actual object size, which depends on focus distance.

Also, the formula assumes ideal optics. Real lenses have aberrations that reduce effective sharpness, and the actual magnification may vary slightly with focus distance for internal focusing lenses. But for most practical purposes, the simple ratio is accurate to within a few percent.

Frequently Asked Questions

What is the difference between magnification and resolving power?

Magnification makes things appear larger, but resolving power is the ability to see fine detail. You can have high magnification but low resolving power if the aperture is small or the optics are poor. The maximum useful magnification is limited by the aperture and the wavelength of light; beyond that, you’re just enlarging blur.

How do I calculate magnification for a camera lens?

For a camera, the “magnification” often refers to the reproduction ratio, which is the image size on the sensor divided by the object size. For a simple fixed-focus lens, you can approximate it as the lens focal length divided by the focus distance (if the distance is much larger than the focal length). But the common use of “magnification” for cameras is zoom ratio, which is the longest focal length divided by the shortest. That’s not the same as the magnification formula here.

Does the eyepiece focal length affect the brightness of the image?

Yes. The exit pupil (the bright circle you see when you hold the eyepiece away) is the objective diameter divided by the magnification. A smaller exit pupil means a dimmer image. For example, a 100 mm telescope at 50× gives a 2 mm exit pupil, which is fine in daylight but dim for night viewing. A 7 mm exit pupil is about the maximum your eye’s pupil can use in the dark.

Can I use this formula for a microscope?

No. A compound microscope has two stages: the objective lens magnifies first, then the eyepiece magnifies that image. The total magnification is the objective magnification (like 40×) multiplied by the eyepiece magnification (like 10×), giving 400×. That’s a different formula from the simple ratio used for telescopes.

What is the maximum useful magnification for my telescope?

A common rule of thumb is 2× per millimeter of aperture (or 50× per inch). So a 70 mm telescope can handle about 140×. Beyond that, the image gets too dim and soft. However, this depends on the quality of the optics and the atmospheric conditions. On a night with poor seeing, even lower magnifications may be unusable.

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in obj | Objective focal length | mm | 500 | 0
in ep | Eyepiece focal length | mm | 25 | 0
out mag | Magnification (×) | × | obj / ep | 1
out ratio | Magnification ratio | :1 | obj / ep | 1