Lens and mirror equations
thin lens equationmirror equationmagnification formula1/f = 1/do + 1/didiopters
The thin lens and mirror equations with the two magnification relations and lens power — everything needed to place an image and say how big it is.
Thin Lens Equation
Relates a thin lens's focal length to its object and image distances.
Focal Length of a Spherical Mirror
A spherical mirror focuses parallel rays at half its radius of curvature.
Lens Magnification (m = −d_i/d_o)
Image magnification from the ratio of image to object distance; the minus sign tracks inversion.
Magnification from Heights (m = h_i/h_o)
Magnification as the ratio of image height to object height.
Lens Power in Diopters
The optician's unit: lens power in diopters is the reciprocal of focal length in meters.
How they fit together
Lenses and mirrors obey the same equation, 1/f = 1/dₒ + 1/dᵢ, which is why one page covers both. Focal length sets the behaviour and the two magnification forms let you cross between geometry and size: m = −dᵢ/dₒ from the distances, m = hᵢ/hₒ from the heights. Lens power in diopters is nothing more than 1/f in metres, which is what an optometrist writes on a prescription.
Almost every error in this topic is a sign error, not an arithmetic one. Converging lenses and concave mirrors take positive f; diverging lenses and convex mirrors take negative f. A positive image distance means a real image on the far side that you could catch on a screen; a negative one means a virtual image on the same side as the object, which is what you see in a magnifying glass held close. A negative magnification means the image is inverted, and |m| > 1 means enlarged. Decide the signs before you touch the calculator, and sanity check the result: an object inside the focal length can never give a real image.