Lens & Mirror Equation Calculator

The Lens & Mirror Equation Explained

The same equation describes how both lenses and mirrors form images, relating the focal length to how far the object and image sit from the lens or mirror.

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$

Magnification: $$m = -\frac{d_i}{d_o}$$

A positive dᵢ means a real image on the opposite side from the object (for a lens) or the same side (for a mirror); a negative dᵢ means a virtual image. The sign of m tells you orientation: positive means upright, negative means inverted, and |m| > 1 means the image is magnified.

Worked Example: A Simple Magnifying Setup

Using the calculator's defaults — a converging lens with f = 10 cm and an object placed at d₀ = 25 cm — solving for image distance gives $\frac{1}{d_i} = \frac{1}{10} - \frac{1}{25} = 0.06$, so $d_i \approx 16.7$ cm. Magnification comes out to $m = -16.7/25 \approx -0.67$, meaning the image is real (positive dᵢ), inverted (negative m), and reduced (|m| < 1) — exactly what you'd expect from a camera lens forming an image on a sensor placed beyond the focal length.

Lenses vs. Mirrors

Both obey the identical mathematical relationship, but the physical picture differs: a lens bends light by refraction as it passes through, while a mirror bends light by reflection off its surface. This calculator applies the same formula to both, since the sign conventions used here are set up so the equation itself doesn't need to change — only how you interpret which side is "real."

A Brief History of the Lens Equation

The mathematical treatment of image formation by lenses developed gradually through the 17th century, with contributions from Johannes Kepler's early work on optics and later refinement by others working on telescope and microscope design. The thin lens equation in its modern form became a standard tool of geometric optics by the 19th century, underpinning the design of cameras, telescopes, and eyeglasses well before anyone could simulate ray paths on a computer.

Common Lens & Mirror Mistakes

Mixing up the sign convention between lenses and mirrors is the most common error — the same numeric result can mean a real image for one and a virtual image for the other, depending on which convention is in use, so it's important to stay consistent with a single convention throughout a problem. Forgetting that magnification's sign carries meaning (not just its magnitude) is another frequent slip, since a negative magnification indicating an inverted image is easy to overlook. Assuming an object exactly at the focal length produces a normal image is a third common mistake — at d₀ = f, the equation gives an infinite image distance, meaning parallel rays that never converge to form an image at all.

Optics Terms You Should Know

Focal Length (f) — the distance from a lens or mirror to its focal point, where parallel rays converge.

Real Image — an image formed where light rays actually converge, projectable onto a screen.

Virtual Image — an image formed where light rays only appear to originate, not projectable onto a screen.

Magnification (m) — the ratio of image height to object height, with sign indicating orientation.

This calculator uses the thin-lens/mirror approximation and a single sign convention; always double-check which convention your textbook or course uses before comparing answers.