Magnification Calculator

Lateral magnification (also called transverse or linear magnification) can be calculated using two equivalent methods:

Method 1 — From distances:

$$M = -\frac{d_i}{d_o}$$

where d_i is the image distance and d_o is the object distance. The negative sign accounts for the sign convention where real images (d_i > 0) are inverted.

Method 2 — From heights:

$$M = \frac{h_i}{h_o}$$

where h_i is the image height and h_o is the object height. A negative h_i indicates an inverted image.

These two definitions are equivalent and arise from the geometry of similar triangles formed by light rays passing through a thin lens or reflecting from a mirror. The sign of M carries important physical information:

• M > 0: Image is upright (same orientation as object) — typically a virtual image
• M < 0: Image is inverted (flipped) — typically a real image
• |M| > 1: Image is magnified (larger than object)
• |M| < 1: Image is diminished (smaller than object)
• |M| = 1: Image is the same size as the object

It is important to distinguish lateral magnification from angular magnification, which is used for instruments like magnifying glasses, microscopes, and telescopes where the relevant quantity is the angle subtended at the eye rather than the physical image size.

For compound optical systems with multiple lenses, the total magnification is the product of individual magnifications: $$M_{total} = M_1 \times M_2 \times M_3 \times \ldots$$

The magnification (M) is the signed value that encodes both size ratio and orientation. A value of M = −2 means the image is inverted and twice the size; M = +0.5 means upright and half the size.

The |M| (Size Ratio) gives the absolute magnification — the pure size ratio without orientation information. The image orientation and image size descriptions provide a plain-language summary of whether the image is upright/inverted and magnified/diminished.