Magnification is one of the simplest ratios in optics, but it shows up in two different-looking formulas depending on what you actually measured — the size of an image, or the distances involved in forming it — plus a third, entirely separate formula for how microscopes and telescopes multiply magnification across two lenses. This article walks through all three, when to use each, and what the sign of the result tells you.
Two ways to measure the same ratio
If you can directly measure or are given the size of an image and the size of the object it represents, magnification is just their ratio: m = image size / object size. A 5 cm image of a 1 cm object is 5x magnification, full stop — no lens properties required. But often you don't have direct size measurements; instead you know how far the object and image are from a lens or mirror. In that case, magnification is the negative ratio of those distances: m = -di/do, where di is the image distance and do is the object distance. Both formulas describe the exact same physical quantity — they just start from different known inputs, which is why this calculator offers a mode switch instead of forcing one formula on every user.
Why the sign matters
In the distance-ratio formula, the minus sign isn't decorative — it encodes orientation. A negative m means the image is inverted (upside down relative to the object), which happens for every real image formed by a single converging lens or concave mirror when the object sits beyond the focal point. A positive m means the image is upright, as with a magnifying glass held close to a page or the virtual image in a flat mirror. The magnitude of m, independent of sign, tells you the size: |m| greater than 1 is enlarged, |m| less than 1 is reduced, and |m| equal to 1 means the image is exactly the same size as the object.
Multiplying magnification across two lenses
A single lens has one magnification, but instruments like compound microscopes and telescopes use two lenses in series — an objective near the specimen and an eyepiece near your eye — and their powers multiply rather than add. A 40x objective doesn't just add 40 units of magnification; it multiplies whatever the eyepiece contributes. That's why a 40x objective with a 10x eyepiece gives 400x total, not 50x. This is a fundamentally different calculation from the size- or distance-ratio formulas above: it's not measuring one image against one object, it's chaining two magnifying stages together, which is why this calculator gives it its own dedicated tab.
Microscopes vs. telescopes
Microscope objectives and eyepieces are conventionally labeled with a power rating (like 40x or 10x) printed right on the barrel, so multiplying them directly is the standard approach and the one this calculator's Microscope/Telescope tab uses. Telescopes, by contrast, are usually specified by focal length in millimeters rather than a bare power rating, and their total magnification is calculated as the objective's focal length divided by the eyepiece's focal length (fo/fe) — a 1,000 mm telescope objective paired with a 10 mm eyepiece gives 100x, following the same underlying principle of two lenses multiplying their effects, just expressed through focal lengths instead of printed power numbers.