Spherical mirror object distance calculator

Enter the focal length and desired image distance in any consistent length unit, using the standard sign convention, and receive the required object distance in that same unit. It works for concave and convex spherical mirrors, identifies combinations that imply an object at infinity, and provides a fast check for classroom optics, laboratory setups, and early optical design calculations.

Enter distances with one consistent sign convention

The calculator uses the common real-is-positive convention for spherical mirrors. Give a concave mirror a positive focal length and a convex mirror a negative focal length. Enter a positive image distance when reflected rays actually converge to form a real image in front of the mirror, or a negative image distance when the image is virtual and appears behind the mirror. Both inputs must use the same physical unit, such as millimeters, centimeters, or meters; because the equation is homogeneous, the result automatically uses that unit too. Do not mix centimeters for focal length with millimeters for image distance unless you convert one first. A sign is part of the measurement, not merely a display preference, so changing it changes the physical arrangement being requested. The calculator accepts decimal values and numeric strings through the API, but both distances must be finite and non-zero. Keeping a quick sketch of the mirror, object side, and image side beside your values is often the simplest way to prevent a sign mistake before calculating.

How the object distance is calculated

A spherical mirror obeys 1/f = 1/do + 1/di, where f is focal length, do is object distance, and di is image distance. Solving specifically for the unknown object distance gives do = f × di / (di − f). This capability applies that rearrangement directly, after validating that the two supplied quantities describe a finite calculation. When image distance equals focal length, the denominator becomes zero. Physically, that limiting case places the object infinitely far away, so there is no finite object distance to return and the calculator reports an input error. Zero focal length and zero image distance are also rejected because the reciprocal form of the mirror equation is undefined there. The result retains its algebraic sign. A positive object distance normally describes a real object in front of the mirror, while a negative result describes a virtual object under the selected convention. Floating-point output is normalized to stable significant precision, making repeated API and browser calculations deterministic while retaining useful accuracy for ordinary optics work.

Interpret the result and check the setup

Treat the returned object distance as the position required by the ideal paraxial mirror model. Its magnitude tells you how far from the mirror vertex to place the object, and its sign tells you which side is implied by the convention. For example, a concave mirror with a positive focal length can produce a positive object distance for many real-image targets. Substituting the returned value into the original equation is a useful independent check: 1/f should equal 1/do + 1/di within normal rounding. The calculation assumes a spherical mirror, rays near the principal axis, negligible mirror thickness, and a focal length that is already known. It does not model spherical aberration, aperture limits, tilted mirrors, wavelength-dependent coatings, or mechanical clearance. Those effects matter in precision systems and should be evaluated with a fuller ray-tracing model. For teaching, bench planning, and quick feasibility checks, however, the ideal equation clearly shows whether the requested image position corresponds to a finite object placement and whether that placement is real or virtual.