Thin Lens & Mirror Equation Calculator
Signed lens and mirror conjugates, image orientation and thin lensmaker power.
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Results are estimates from the formulas shown, not a professional design or certification. Have a qualified engineer verify anything safety-critical (structures, electrical installations, gas or pressure systems).
About the Thin Lens & Mirror Equation Calculator
Solve the signed thin-lens or spherical-mirror equation for focal length, object distance or image distance. Results include transverse magnification, real or virtual image classification, orientation and reciprocal focal length. The drawing keeps outgoing mirror rays on the incident side and lens rays on the transmitted side; dashed lines are virtual extensions.
The free preview masks calculated figures and marks the diagram. A Pro pass opens the complete result and reports. A separate thin-lensmaker model accepts signed surface curvatures and the refractive indices of the lens and its surrounding medium. This is an ideal paraxial estimate, not an optical prescription.
How to use it
- Choose conjugates or the thin-lensmaker model.
- For conjugates, choose lens or mirror and the one unknown; enter the other signed distances in the selected common unit.
- Use positive focal length for a converging lens or concave mirror, and negative for a diverging lens or convex mirror.
- For lensmaker, enter both indices and signed radii for light travelling left to right, then inspect the assumptions.
Examples
f = 20 cm, real object distance 30 cm
Image distance 60 cm; magnification -2; real inverted image
f = -20 cm, real object distance 30 cm
Image distance -12 cm; magnification 0.4; virtual upright image
Lens n = 1.5 in medium n = 1, R1 = 20 cm, R2 = -20 cm
Focal length 20 cm; reciprocal focal length 5 m⁻¹
f = 20 cm, object distance 20 cm
Ideal image at infinity; no finite transverse magnification
Signed convention
1/f = 1/do + 1/di and m = -di/do. A real object on the incident side has positive do; a virtual object has negative do. A real image has positive di and a virtual image negative di. A converging lens or concave mirror has positive f; a diverging lens or convex mirror has negative f.
For a lens, real outgoing rays travel right, while for a mirror they return left. Virtual extensions do not indicate light travelling backwards. The drawing uses independent transverse and longitudinal scales and clips rays at the viewport without changing their directions.
Thin lensmaker and power
1/f = (n_lens/n_medium - 1)(1/R1 - 1/R2). A curvature centre to the right is positive for left-to-right incident light. Thickness terms are omitted. In metres, reciprocal focal length has units m⁻¹, called dioptres for lens power. Equal relative indices or equal reciprocal curvatures give zero power and an ideal infinite focal length.
Sources
The paraxial models, sign conventions and lensmaker equation follow OpenStax: spherical mirrors and OpenStax: thin lenses.
Limitations
- Thin elements, small angles and negligible aberrations are assumed.
- Finite nonzero known distances are required; an exactly zero reciprocal can yield an ideal infinite solved distance.
- No thickness, aberration, aperture, image-brightness or multi-element system calculation is included.
- The result is an estimate, not a professional optical design or eye prescription.
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Frequently asked questions
What do I get without a pass?
Without a pass, Thin Lens & Mirror Equation Calculator shows the rules, dates and checks that apply to your figures, with every computed amount hidden. Until you unlock it, the result can’t be downloaded or copied. A Pro, Premium or Ultimate pass, a one-time payment that never renews, unlocks the full result. The pricing page lists the passes and their prices.
Where does a virtual image appear?
A virtual lens image lies on the incident side; a virtual mirror image lies behind the mirror. Its signed image distance is negative in the convention used here.
What does a negative magnification mean?
It indicates inversion relative to the signed object height. A positive magnification indicates an upright image. At an infinite image distance there is no finite transverse magnification.
Why does the lensmaker equation need the surrounding index?
The thin-lens power depends on the ratio of the lens index to the medium index. Treating an immersed lens as though it were in air can give the wrong focal length.
Why does the focal-point case give infinity?
When the conjugate's reciprocal is exactly zero, ideal rays leave parallel and there is no finite image plane. The calculator states that limit instead of inventing a finite distance.