About the Magnification of a Lens Calculator
This calculator uses the thin lens equation to find where a lens forms an image and how large that image is relative to the object. Enter the lens's focal length, how far the object sits from the lens, and the object's height, and the tool returns the image distance, the magnification, the image height, and a plain-language description of the image: real or virtual, upright or inverted, enlarged or reduced.
The thin lens equation
For a thin lens, the object distance d₀, the image distance dᵢ, and the focal length f are related by:
1/f = 1/d₀ + 1/dᵢ
Solving for the image distance gives dᵢ = (f × d₀) / (d₀ − f). The lateral magnification compares image size to object size:
m = −dᵢ / d₀ = hᵢ / h₀
where hᵢ is the image height and h₀ is the object height. Multiplying the magnification by the object height gives the image height directly: hᵢ = m × h₀.
Reading the sign convention
- Focal length f: positive for a converging (convex) lens, negative for a diverging (concave) lens.
- Object distance d₀: always entered as a positive value — the object is assumed to sit in front of the lens, on the side the light comes from.
- Image distance dᵢ: positive means a real image forms on the opposite side of the lens from the object (light rays actually converge there); negative means a virtual image forms on the same side as the object (rays only appear to diverge from it).
- Magnification m: negative means the image is inverted relative to the object; positive means it is upright. When |m| > 1 the image is enlarged, when |m| < 1 it is reduced, and when |m| = 1 it is the same size as the object.
A worked example
A converging lens with f = 10 cm holds an object of height 5 cm at d₀ = 30 cm. The image distance is dᵢ = (10 × 30) / (30 − 10) = 15 cm, a positive value, so the image is real and forms 15 cm on the far side of the lens. Magnification is m = −15/30 = −0.5, so the image is inverted and reduced to half size: hᵢ = −0.5 × 5 = −2.5 cm, meaning a 2.5 cm tall image standing upside down.
Special case: object at the focal point
If the object sits exactly at the focal length (d₀ = f), the denominator d₀ − f becomes zero and the lens equation has no finite solution — the outgoing rays leave the lens parallel and the image forms at infinity. The calculator flags this case instead of returning a number.
Where the thin lens model applies
This is the standard single-element, paraxial (small-angle) thin lens formula used throughout introductory and intermediate optics. It assumes a lens whose thickness is negligible compared to its focal length and object/image distances, rays close to the central axis, and no aberration. It is the right tool for single-lens problems in photography, projector setups, simple telescopes, and classroom optics. It is not exact for thick lenses, multi-element camera zoom lenses, or compound microscopes, where each element must be modeled in sequence and the results combined.