Using the model
Symbols, assumptions and limitations
The calculator uses the real-is-positive sign convention shared by the displayed thin-lens and spherical-mirror equations. Positive focal length describes a converging lens or concave mirror; negative focal length describes a diverging lens or convex mirror. Real object and image distances are positive, while virtual object and image distances are negative.
Paraxial (Gaussian) approximation: the rays are assumed to travel close to the optical axis and make small angles with it, so sin θ ≈ tan θ ≈ θ when θ is measured in radians. This first-order approximation makes the thin-lens equation possible. Rays far from the axis, wide apertures, and steep angles reveal spherical and other aberrations and need a more complete ray-tracing model.
Worked examples
See the method in practice
Converging system
With f = 10 cm and dₒ = 30 cm, the image forms at dᵢ = 15 cm with m = −0.5: real, inverted, and reduced.
Object inside focal length
With f = 10 cm and dₒ = 5 cm, dᵢ = −10 cm and m = +2: virtual, upright, and enlarged.
Questions
Frequently asked
Which sign convention is used?
The calculator uses real-is-positive signs: dₒ is positive for a real object and negative for a virtual object, dᵢ is positive for a real image and negative for a virtual image, and f is positive for converging systems.
What does negative magnification mean?
The image is inverted relative to the object. Positive magnification means it is upright.
What happens when dₒ equals f?
Emerging rays are parallel and the ideal image lies at infinity, so no finite image distance is returned.
Sources & review
Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.
OpenStax University Physics — Thin Lenses OpenStax University Physics — Spherical MirrorsWritten by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology