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Mirror Equation Calculator

The mirror equation calculator helps you find the unknown among the object position, image position, and focal length of a mirror.

Clear
Image distance166.6667 mm166.667 mm behind the lens — a real image you could catch on a screen
Magnification-0.66667×real, inverted, and reduced to 0.667×
Image height13.3333 mm13.333 mm, inverted
Focal length100 mmconverging
Optical power10 dioptresthe reciprocal of the focal length in metres — how spectacle prescriptions are written
Object distance in focal lengths2.5fbeyond 2f — a real, inverted, reduced image, as in a camera
Focal length from the surfaces115.3846 mm(n−1)(1/R₁ − 1/R₂) with n = 1.52 and a flat second face
Power from the surfaces8.6667 dioptres

The formula

1/f = 1/d_o + 1/d_i; m = −d_i ÷ d_o; D = 1 ÷ f

Signs carry the physics

The thin-lens equation is simple arithmetic; the conventions are where errors live. A positive image distance means a real image, formed behind the lens, which you could catch on a screen. A negative one means a virtual image on the same side as the object, visible only by looking through the lens. A negative magnification means inverted.

Where the object sits relative to the focal length determines the character of the image entirely. Beyond 2f you get the reduced inverted image a camera uses. Between f and 2f you get a magnified inverted one, which is a projector. Inside f the image turns virtual, upright and magnified — a magnifying glass. Exactly at f the rays emerge parallel and the image goes to infinity, which is what a collimator exploits.

Dioptres are simply the reciprocal of focal length in metres, and they add: two thin lenses in contact have the sum of their powers. That additivity is why prescriptions are written this way rather than in millimetres of focal length.

Vertex distance is a real clinical correction. Moving a lens closer to the eye changes its effective power, and the shift grows with the prescription — negligible below about 4 dioptres, and several tenths of a dioptre for strong corrections, which is why contact lens and spectacle prescriptions genuinely differ.